Call Level Performance Analysis for Multiservices Wireless Cellular Networks With Adaptive Resource...

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 4, JULY 2005 1455

Call Level Performance Analysis for MultiservicesWireless Cellular Networks With Adaptive Resource

Allocation StrategiesLauro Ortigoza-Guerrero, Felipe A. Cruz-Perez, Member, IEEE, and Heraclio Heredia-Ureta

Abstract—In this paper, a novel mathematical approach to eval-uate the performance of adaptive or flexible resource allocation(FRA) strategies with uniform quality of service (QoS) provision-ing in multiservices wireless cellular networks, in terms of the callblocking probabilities and transmission delay, is proposed. FRAstrategies improve channel utilization by dynamically adjustinguser transmission rates. Based on the available cell capacity, thebandwidth offered to users can be adjusted in accordance with theelasticities of their service types. When an FRA strategy providessimilar QoS to all the calls of the same service type, it is knownas an FRA with uniform QoS provisioning. In multiservices wire-less cellular networks, calls can be identified as belonging to oneof four service classes (i.e., conversational, streaming, interactive,and background). Transmission delay is one of the most importantperformance measurements of interactive and background serviceclasses. Transmission delay, however, has not been addressed inprevious studies on FRA with uniform QoS provisioning either inconjunction with interactive or background service classes. Thisis because such studies have been based on the Markov propertyof the negative exponential distribution of the service time whichlacks time delay information. The analytical approach in this paperis based on the fact that in FRA strategies with uniform QoS pro-visioning, calls of all service types tend to use an average numberof resources (i.e., the number of resources available for a servicetype divided by the number of active users of that service type).This feature facilitates the assessment of the mean service time andthe number of resources allocated to a call during its lifetime and,consequently, a call’s transmission delay. The study also considersthe fact that the probability density function (pdf) of the normal-ized transmission delay is almost a symmetrical function and haslow variance. The accuracy of the proposed mathematical analysisis then corroborated by means of semianalytical methods and bydiscrete event computer simulation.

Index Terms—Adaptive resource allocation, call level perfor-mance analysis, integrated services, resource management, unen-cumbered service time, wireless cellular networks.

I. INTRODUCTION

THE 2.5 and 3G networks will provide new and improvedtelecommunication services to mobile users. They will

offer, in a single terminal, a range of different services that canbe separated into four different classes [1]: a) conversational,b) streaming, c) interactive, and d) background. The primaryfactor distinguishing these four classes is how delay-sensitive

Manuscript received November 26, 2003; revised June 5, 2004 and September23, 2004. The review of this paper was coordinated by Prof. Y.-B. Lin.

L. Ortigoza-Guerrero is with Wireless Facilities, Inc., San Diego, CA, USA.F. A. Cruz-Perez is with CINVESTAV-IPN, Mexico City, Mexico.H. Heredia-Ureta is with Universidad de Occidente Campus Culiacan.Digital Object Identifier 10.1109/TVT.2005.851306

the traffic is. The conversational class is meant for very delay-sensitive traffic, while the background class is the most delay-insensitive. Each service class consists of multiple applicationswith different quality of service (QoS) requirements such asvoice, video, videophone, fax, ftp, web browsing, etc.

The conversational class provides high quality access to arange of different services including high rate services. Thisclass is suitable for demanding applications that require band-width guarantees. The streaming class is designed to carry highbandwidth, variable-bit rate services, such as a medium or highquality video- or teleconferencing service. The holding timefor these two classes is independent of the actual throughput re-ceived during the service time. The interactive class supports lessdemanding services typically supported by today’s best effortIP networks, including file transfer, web browsing, or telnet ap-plications. UMTS networks are expected to provide some formof throughput guarantee even for these types of service [2]. Theservice or transmission time of the interactive calls1 typicallydepends on the throughput (the transfer of a file, for instance,would take half of the time with doubled throughput). The back-ground class belongs to the best effort types of service. That is,background calls are allocated whatever bandwidth is left overby calls carrying other service classes with higher priority. Thereis no minimum rate guarantee for the background class. Exam-ples of this class include e-mail and low quality file transfers.This class is similar to the interactive class with respect to thetransmission time.

The effect that the variation of the number of resources (band-width) granted in each session has on the service time, therefore,depends on the class of service the session belongs to. In the con-versational as well as in the streaming class of services, servicetime is not modified directly by the allocation of variable band-width, but may be modified by the subjective QoS evaluation thefinal user makes [3]. In contrast, there are applications wherethe service time directly depends on the amount of resourcesgranted to a particular session (i.e., fax, data file transfer, etc.).

A service’s bit rate can be controlled adaptively by the ra-dio access network depending on the air interface loading andthe requested QoS (this is known as adaptive multimedia inwireless/mobile networks [3]–[11]). Low bit rates are offeredduring high loading to achieve higher capacity while providingslightly lower QoS. The opposite is also true. A service’s bitrate can also be modified when a class-based QoS over air in-terface exists, as proposed in [12]. With this approach, the bit

1The terms “session” and “call” are used indistinctly to refer to data connec-tions of the interactive and background classes.

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rate and/or priority depends on the various subscription typesavailable (i.e., Premium, Gold, and Silver). For conversationalclass of service like speech, for instance, Premium speech, Goldspeech, and Silver speech could be available. Each combination(subscription type + example of service class) offers character-istic performance to its customers. In this paper, service typeis defined as a combination of a specific service with a sub-scription type (i.e., Voice-Premium, Video-Gold, etc.). Each ofthese different service types will demand a different number ofnetwork requirements.

2.5G and 3G networks will need to use resource aggregationin order to cope with high data rates. This approach, however,produces a high blocking rate because few users are assignedmultiple resources. In order to improve total channel utilization,several flexible resource allocation (FRA) strategies have beenproposed in the literature [3]–[16]. These strategies selectivelyadjust the user transmission rates according to network condi-tions. Based on the available capacity of a cell, the bandwidthoffered to users is adjusted in accordance with the elasticitiy2 oftheir service types (adaptive multimedia services and subscrip-tion types).

The amount of bandwidth granted to a call or session depends,among other things, on the available bandwidth, the interferencelevel, the subscription type, and the service type the call iscarrying. The aim of a flexible (or adaptive) resource allocationstrategy is to find a tradeoff between the existing capacity andthe QoS. When an FRA strategy provides similar or fair QoS toall the calls of the same service type, we refer to it as a uniform3

QoS provisioning FRA strategy. Uniform QoS provisioning, orintra-fairness, is achieved by equally or fairly sharing resourcesamong users with the same service types.4

There are several examples of FRA strategies previously pro-posed in the literature5 for circuit switched multiservice wire-less cellular networks [5]–[9], [11], [12], [17], [18], [20]–[22].In [7], an adaptive QoS handoff priority scheme that makes thebest of the ability of most streaming multimedia traffic typesto adapt to, and to trade off, QoS with changes in the amountof bandwidth was proposed. In [20], the authors proposed athreshold-type call admission control (CAC) algorithm for QoSprovisioning in a system withK classes of adaptive multimedia

2The bandwidth users are allocated ranges from a minimum required value toa maximum requested value (which is referred to as throughput window in [17]).

3In [18], two kinds of “fairness” are distinguished: inter-fairness (the fairnessamong traffic classes) and intra-fairness (the fairness within one traffic class).Intra-fairness is equivalent to the uniform QoS provisioning term used here andthe intra-class fairness term used in [19].

4As stated in [19], due to bandwidth adaptation, calls of the same class mayoperate at different bandwidth levels and this is undesirable from the usersperspective. Thus, QoS provisioning schemes should be fair to all calls withinone class and should be considered as another QoS constraint in the FRAstrategies.

5The strategies H2 and S1 proposed originally in [16] for cellular systemswith a single service type can be easily extended for multi-services wirelesscellular networks. H2 is not an FRA but provides the upper bound for FRAsin terms of capacity. It always allocates the minimum bandwidth requested. S1allocates the maximum bandwidth requested unless available resources are notenough. S1 is partially flexible because it does not allow a “fine adjustment”of the number of resources assigned to active calls and cannot provide uniformQoS to users. S1 does not provide uniform QoS because it allocates differentbandwidth to calls of the same service type. The generalized versions of thesestrategies are evaluated in this paper for comparative proposes.

services sorted according to their priority. In [18], a bandwidthallocation algorithm for multiple classes of users with fairnessamong classes and fairness within one class in adaptive mul-timedia services (layered coding assumed) in wireless/mobilenetworks was studied. In [6] and [17], a generalization of thestrategy equal resource sharing allocation (ERSA) [14] for mul-tiservices cellular systems was proposed. ERSA equally sharesresources among the different service types based on the num-ber of ongoing calls in a cell through the use of resource re-assignments. In [5], an FRA strategy called flexible resourceallocation with differentiated priorities and QoS (FRAQoS) wasproposed. FRAQoS copes with multiple service types and meetsthe upper capacity limit of FRA strategies. At the same time, [5]also introduces the concepts of prioritized call degradation andprioritized call compensation. Both ERSA and FRAQoS are uni-form QoS provisioning FRA strategies. However, all the above-mentioned works have considered only conversational and/orstreaming service classes (i.e., voice, audio phone, streamingvideo, videophone, videoconference, etc.).

On the other hand, most of works considering adaptive re-source allocation for interactive and/or background serviceclasses (i.e., interactive multimedia, web browsing, video ondemand, e-mail, paging, fax, remote login, data on demand,file transfer, retrieval service, SMS, download of databases, re-ception of measurement records, etc.) have evaluated the calllevel performance [7], [11], [23] through discrete event com-puter simulation [8], [9], [22]. In fact, a literature survey showsthat relatively few in depth papers have been published on themathematical analysis of the call level performance of FRA con-sidering service applications whose unencumbered service timedepends on the allocated bandwidth [11], [15], [24]. In [15], anFRA strategy is presented in a mobile GSM cellular networkwith two service types: voice, and high speed circuit switcheddata (HSCSD). The strategy of [15] assigns as many resources orbandwidth as possible to incoming HSCSD calls. If the numberof resources (bandwidth) assigned falls between the minimumand maximum required by the request, the HSCSD call eitheris queued (provided there is still room in a waiting queue) orblocked and cleared from the system. A multicell analysis isperformed allowing a call to reduce or increase its service rateat each handover occurrence assuming that the unencumberedservice duration of a HSCSD call is a function of the numberof channels granted to an individual call. This strategy does notprovide uniform QoS. This strategy is mathematically analyzedby means of a multidimensional Markov chain, where one statevariable is used for each possible number of allocated resources.However, this method cannot be used to analyze the performanceof FRAs where degradation/compensation techniques are usedto adapt to changing network bandwidth availability because oftheir dynamic nature.

In [24], a call level model of UMTS core networks is devel-oped (single transmission link modeling) where calls belongingto one of the four UMTS service classes equally share the linkcapacity. The actual holding time of the interactive class callsare determined using the Markov model of the transmissionlink based on the Markov property of the negative exponentialdistribution of the service time. The performance of complete

ORTIGOZA-GUERRERO et al.: CALL LEVEL PERFORMANCE ANALYSIS FOR MULTISERVICES WIRELESS CELLULAR NETWORKS 1457

resource sharing and complete resource partitioning strategiesis then evaluated in terms of the blocking probabilities andthe class-wise throughput. The methodology and assumptionsof [24] prevent the calculation of the transmission delay of in-dividual sessions, however, since time delay information is lostdue to the memoryless property consideration.

Finally, in [11], an FRA strategy with complete resource shar-ing, dual threshold bandwidth reservation (DTBR), and equalresource sharing for wireless cellular networks with voice andinteractive class data services is mathematically analyzed. Themethodology of [11] is also based on the Markov property of theexponential distribution of the call duration times or call holdingtimes. The authors assume that data calls are serviced with an op-timum departure rate if they are provided with maximum band-width, and, due to the Markov property, the real instantaneousdeparture rate of data calls is proportional to the actual allocatedbandwidth of each call. Again, with the mathematical analysisof [11], the transmission delay cannot be calculated since thetime delay information is lost due to the memoryless propertyconsideration. Additionally, the analysis methodology of [11] islimited to call duration times with exponential distribution.

Thus, there is currently no mathematical analysis to evalu-ate the transmission delay of FRA strategies with uniform QoSprovisioning in wireless cellular systems with services whosesession service times depend on the allocated bandwidth (i.e., in-teractive and background service classes). In this paper, a novelalgorithm to mathematically analyze the performance of suchFRA strategies is presented. The proposed analysis is based onthe fact that in FRA strategies with uniform QoS provisioning,all the active sessions of each service type equally share theleftover resources and tend to utilize an average bandwidth. Themathematical analysis is formally justified and validated by asemianalytical method. The proposed approach allows for theevaluation of the transmission delay for both the interactive andbackground service classes. Furthermore (and contrary to theanalysis of [11]), when coordinate convex access policies6 areused [25], [26], the analytical approach proposed in this pa-per is applicable to unencumbered service times with arbitraryprobability distribution as well.

This paper is organized as follows: Section II describes thesystem model considered in this work. Section III explains tworepresentative FRA strategies with uniform QoS provisioningconsidered as case studies. In Section IV, the effect of the al-located bandwidth on the transmission time in different FRAstrategies is analyzed. The proposed analysis is described inSection V. Finally, numerical results and conclusions are shownin Sections VI and VII, respectively.

II. SYSTEM MODEL

A multicellular homogenous system with multiple conver-sational, streaming, and interactive circuit switched services isassumed where all cells are statistically identical. Hence theoverall system performance can be analyzed by focusing on

6In coordinate convex access policies, the state probabilities can be decom-posed into a simple product form [25], [26]. In such conditions, the solution ofthe steady state balance equations can be expressed in product form.

a single given cell. The general guidelines of the models pre-sented in [11] and [13] are adopted to analyze the FRA strategiesproposed here. These guidelines have been widely used and ac-cepted in the literature, and allow the strategies presented hereto be cast in the framework of multidimensional birth and deathprocesses [5], [6], [10], [11], [13], [17], [27].

Conversational and streaming service class calls are charac-terized by their minimum and maximum bandwidth require-ment, call arrival rate, and departure rate. Although the band-width occupied by conversational and streaming calls mayfluctuate (i.e., half-rate and full-rate speech coding7 in GSM[28]–[30], adaptive multi-rate (AMR)8 speech coding in UMTS[31]–[33], scalable video coding9 [35], [36], etc.), their actualtransmission time is not influenced by the received throughputthroughout their residence in the system. In the conversational,as well as in the streaming classes of service, transmission timeis not modified directly by the allocation of variable bandwidth(throughput) but by the subjective QoS evaluation the final usermakes.

Interactive class calls are characterized by their peak andminimum bandwidth requirements, call arrival rate, and their“ideal” departure rate [24]. The ideal departure rate is experi-enced when the peak bandwidth is used. The actual instanta-neous departure rate is proportional to the bandwidth allocated.Thus, when the bandwidth allocated to an interactive call dropsbelow its peak bandwidth requirement, the actual transmissiontime of the call increases. As such, the transmission time of callscarrying interactive services depends not only on the amount ofdata to transmit (which is a random variable) but also on theallocated bandwidth the calls are granted during their holdingtime. As in [11], it is assumed that each data connection is al-ways provided with the maximum requested bandwidth if thereare enough available channels. When this is not feasible, datacalls of the same service type equally share the leftover band-width. This data traffic model may cover the interactive serviceclass defined in UMTS/IMT2000 [37] and the background ser-vice class if the minimum required bandwidth is set near zero.A similar data traffic model was adopted by call admissionschemes in [5], [6], [9], [38]. The system model includes thefollowing considerations.

1) In the description of the FRA used, Na represents thenumber of idle resources in the cell. As in previous re-lated works [4]–[17], [27], [39], a fixed capacity per cell

7GSM specifications describe two channel coding modes: full-rate (FR) andhalf-rate (HR) [28], [29]. When a traffic channel is in HR mode, one timeslot,which normally serves one connection in FR mode, may be shared by twoconnections, thus doubling the number of connections that can be handled bya transceiver. Thus, the HR feature allows operators to accommodate moreusers with the same hardware resources but at the expense of slight call-qualityreduction.

8In the AMR technique in UMTS [31]–[33], the multi-rate speech coder is asingle integrated speech codec with eight source full-rates: 12.2 (GSM-EFR),10.2, 7.95, 7.40 (IS-641), 6.70 (PDC-EFR), 5.90, 5.15, and 4.75 kbps.

9In the scalable video coding technique (i.e., MPEG-2 [34]), a video sequenceis compressed into several layers: a base layer and several enhancement layers[35], [36]. The base layer can be independently decoded and it provides basicvideo quality; the enhanced layers can only be decoded together with the baselayer and serve to further refine the quality of the base layer. As such, both theframe rate for transmission and the encoded data rate per video frame can bereduced if channel resources become scarce.


is considered here. Each cell has a maximum number ofresources (Nc) with neither guarded resources for handoffcalls or new calls. The adoption of this assumption is jus-tified, since the primary objective of this work is to studythe basic performance of the strategies presented. Con-sidering a channel reservation scheme for handoff callswould bias the performance of each of the strategies andwould not allow for the assessment of their “pure” perfor-mance. However, the combination of the FRA strategiesproposed with call admission prioritization schemes is de-sirable and can be easily achieved.10 The fact that no re-served resources for handoffs are considered in this workallows for the use of coordinate convex access policiesand the decomposition of state probabilities into a simpleproduct form. Furthermore, the mathematical analysis isapplicable to unencumbered service times with arbitrarydistribution [40].

2) In each cell, there are z different circuit switched services(which may be conversational or streaming, or interactiveclasses) (1, 2, . . . , z).

3) A QoS framework based on the paradigm of subscrip-tion type is adopted as suggested in [12]. It is assumedthat the service j can have Cj different subscription typesassociated (i.e., Premium, Gold, Silver, etc.). That is, sev-eral subscription types can exist for each service. Eachsubscription type for a specific service offers characteris-tic performance to its customers. For example, Premiumsubscription associated to voice service can offer the ne-gotiated bandwidth at all times, regardless of congestionon the air interface. Each of the remaining subscriptiontypes has a certain elasticity associated with it, where theelasticity of the Gold subscription is more than that of thePremium subscription but less than that of the Silver. Incase of congestion on the air interface, bandwidths offeredto users are adjusted in accordance with the elasticities oftheir classes so that congestion is mitigated. Note that eachof the z conversational services can have different band-width and/or elasticity requirements (i.e., different QoS).

4) Service type is defined as the combination of a specificservice with a subscriber type (e.g., Voice-Premium).11

Thus, since the system has z services, each with Cj sub-scription types, then the system has n = C1 + · · · + Cz

service types. Calls of service type imay require a numberof resources between a maximum mMi and a minimummmi. Thus, the greater the difference mMi −mmi, thegreater the elasticity of the service type i. This elasticity,combined with the bandwidth requirements, renders char-acteristic (qualitative) performance to each service type.Mobile users can associate their applications with the ap-propriate subscription type based on their QoS expectation

10In this work, we deal with prioritized resource sharing among the differentservice types instead of prioritized admission control.

11In summary, “service class” refers to a conversational, streaming, inter-active, or background service classes defined by UMTS. “Subscription type”refers, for example, to Premium, Gold, and Silver. Service refers to video, voice,data. Finally, “service type” is defined as a combination of a specific servicewith a subscription type (i.e., Voice-Premium, Video-Gold, etc.).

and the pricing scheme. As explained in [12], subscriptiontypes can be preconfigured with user’s applications, or ex-plicitly selected at the time the application starts. Mobileusers can make decisions as to whether to stay with thatsubscription type or switch to a higher (lower) type, basedon perception of QoS and the pricing scheme. Using thisQoS framework, service negotiation between mobile usersand network operators can be quite flexible.

5) The new call arrival process of each service type follows aPoisson process. The mean new call arrival rates for eachservice type are λ1, λ2, . . . , λn. It is assumed that the newcall arrival rate produced by calls requesting the servicetype i (for i = 1, . . . , n), λi, is a fraction, fi, of the totalnew call arrival rate.

6) The handoff call arrival process for each service typedue to the motion of users is also considered to bePoisson distributed with mean handoff arrival ratesλh1, λh2, . . . , λhn, respectively for each service type.These handoff arrival rates are calculated by the iterationmethod described in [13] (i.e., the Erlang fixed point ap-proach). In the evaluation environment it is assumed thateach cell receives handoff calls from six different cells.The handoff call arrival process generated by a single cellis clearly not Poisson. However, the combined processfrom the six different neighboring cells can be adequatelyapproximated by a Poisson process [41]–[43]. [41] con-cluded that the assumption of Poisson handoff arrivalsis reasonable. [42] provides additional evidence that thePoisson handoff arrival model can be used with confi-dence. Furthermore, the authors of [43] concluded thatboth Poisson and Modulated Markov Poisson Process(MMPP) models are suitable to represent call arrival pro-cesses in cellular systems, and that both models makethe analysis of such systems tractable. These works alsoconcluded that the Poisson assumption yields the sim-plest analysis: even when exact quantitative results maybe slightly different relative to those obtained with thePoisson assumption, qualitative results and general trendsremain unchanged [42], [43].

7) The (ideal)12 unencumbered service duration of a ses-sion has an arbitrary probability distribution with mean1/µMax1, 1/µMax2, . . . , 1/µMaxn, for service types 1 ton, respectively. The unencumbered call duration of a con-versational or streaming class call does not depend on theallocated bandwidth (or the number of resources) assignedto the call.13 This is not valid for interactive class servicesas their unencumbered call duration does depend on theallocated bandwidth.

8) The cell dwell time (the time spent by a mobile station ina cell independent of being engaged in a call) is a negative

12For interactive class data sessions, the ideal unencumbered service time isexperienced when the peak bandwidth is allocated along the entire duration ofthe session.

13If the number of resources allocated increases, then the bit error rate (BER)decreases but the unencumbered call duration remains practically constant. Theextra number of resources is used to increase redundancy rather than to increasethroughput.


exponential distributed random variable. The cell dwelltime for calls with service type i has a mean equal to 1/ηi.That is, it is considered that each service type is associatedwith a mobility type. It may appear that the negative ex-ponential distribution assumption is inadequate since celldwell time tends to behave more like a gamma distribu-tion [44]; consequently, a general distribution should beused in the model. In such a case, however, analytical so-lutions are neither readily available nor easy to obtain. Theexponential assumption is, therefore, a good performanceapproximation. Essentially, only the average cell dwelltime matters. When the average cell dwell time is smallcompared to the call duration, there is no expected differ-ence between the exponential assumption and the gammadistribution. When cell dwell times are large, the differ-ence becomes more noticeable, but the exponential as-sumption indicates general performance trends [45]. Thisjustifies the use of the exponential assumption to conductthe study.

9) The teletraffic analysis presented here is applicable to acircuit switched TDMA-based wireless system. Neverthe-less, the general concepts are not restricted to systems withthis multiple access scheme and can also manage circuit-switched services in CDMA networks. As in most of thereferences listed, circuit or virtual circuit switching is usedso that the system operates by reserving some communi-cations resources for any call (session) in progress. Asin [39], the network is assumed to use connection orientedresource allocation to provide varying levels of QoS, andthe teletraffic analysis here presented applies to work beingdone in many areas (MPLS, ATM, TCP/IP/RSVP, etc.).The analysis does not assume, however, that connectionoriented mechanisms are used throughout the network. Itis only assumed that connection oriented mechanisms areused in access networks to limit the number of connectionsthat inject traffic into backbone networks.

III. FRA STRATEGIES WITH UNIFORM QoS PROVISIONING

An FRA strategy with uniform QoS provides, on average,the same QoS to the different ongoing calls of a given servicetype. Several FRA strategies with uniform QoS provisioninghave been proposed in the literature. Examples of them areERSA [6], [17], FRAQoS [5], ERSAQoS [6], and DTBR [11].The first three strategies meet the upper capacity limit of FRAstrategies and (with the exception of ERSA) allow the prioritiza-tion of particular services types through the use of prioritizationtechniques. In this work both ERSA and FRAQoS are consid-ered since they are the most representative strategies of theirkind. A brief description of each strategy is given below: ERSAis described in Section III-A and FRAQoS in Section III-B. Sec-tion III-C presents a teletraffic analysis for FRA strategies withuniform QoS provisioning but it is restricted to those strategieswhich provide maximum capacity. In the FRA strategies withmaximum capacity, complete resource sharing is used and, ifnecessary, all active calls can be degraded until its minimumrequired bandwidth.

A. The ERSA Strategy

The equal resource sharing allocation (ERSA) strategy wasoriginally proposed in [14], taking a single service type intoaccount. This strategy equally distributes cell resources amongactive users by modifying the number of allocated resources toactive calls through the use of resource reallocations, triggeredby call arrivals and call departures. A generalization of ERSAwas presented in [6] and [17] for the case where up to n differentservice types with particular bandwidth requirements exist. Theresulting strategy is still referred to as ERSA and it is describedbelow.

1) Allocation of Resources: If a new call or handoff attemptof service type i arrives at a given cell, the following may occur:

a) IfNa ≥ mMi, then the call is assignedmMi resources. Inthis case, all of the active calls in the cell of any servicetype are served with mMi resources.

b) IfNa < mMi, then degradation of ongoing calls in the cellcould be carried out with the aim of releasing resourcesand assigning the incoming call at least mmi resources.If, after call degradations are executed, Na ≥ mmi, thenthe incoming call is assigned at leastmmi resources. Oth-erwise, the call is blocked or dropped.

Call degradations in ERSA are used so that all active calls(including the incoming ones) of any service type experiencethe same QoS. If reshuffling of resources does not provide theincoming and the other active calls with at least the minimumrequired resources, then the incoming call is blocked.

As noted, ERSA assumes there are n different service types.The reshuffling of resources consists of two phases. The firststep aims at distributing theNc resources among the n differentgroups of calls, ensuring that the QoS experienced by each groupof users is the same. The second step equally distributes theresources available among all active calls within each particulargroup.

a) Distribution of Resources Among Different ServiceTypes: A dynamic resource distribution among the n differ-ent groups of active calls is carried out with the final aim ofproviding all active calls (including the incoming call) a set ofresources that allows them to experience the same QoS, irre-spective of the service type they have. To provide similar QoSto calls with different service types, it is assumed that the QoS ofan active call is directly proportional to the number of resourcesit has been allocated (although this is not necessarily true andanother assumption can easily be made). Hence, a user withservice type i andmMi allocated resources has the same QoS asa user with service type j(j �= i) andmMj allocated resources.A user with service type i and mmi allocated resources has thesame QoS as a user with service type j(j �= i) and mmj re-sources. The reshuffling of resources is carried out as explainedbelow.

Each of the n different groups of active calls (each consistingof calls with the same service type) is dynamically assigneda number of resources. Group i is assigned wi resources tobe shared among all active calls with service type i. The vectorw = {w1, w2, . . . , wn} represents, therefore, the distribution ofresources among the different service types. The vector w may


have different combinations, but must fulfill the conditionn∑

t=1

wt = Nc (1)

To ensure the same QoS is provided to all active calls, theexact number of resources assigned to the group iwith ki activecalls (with the type of service i), v i, should be

vi = [mmiki + p(mMi −mmi)ki]; for 1 ≤ i ≤ n (2)

The number of allocated resources to the group of active userswith service type i (group i), vi, must be proportional to the num-ber of active users with that service type, ki. Each active userwith service type imust be assigned at least the minimum num-ber of resources required mmi. This is represented by the firstterm within brackets. The surplus of resources (those extra re-sources not necessary to provide the minimum quality required),Nc-(mm1k1 + · · · +mmnkn), must be distributed among theactive calls of the different service types. The number of surplusresources assigned to each service type depends on the dynamicrange of the resources required by the service type (i.e., mMi-mmi), and the number of active users with each service type inthe cell, (i.e., ki). p is an intermediate variable that determinesthe resource distribution among the different service types. passumes a value between 0 and 1 (0 ≤ p < 1) so that the totalnumber of allocated resources among the n different servicetypes is equal to Nc. That is,

n∑t=1

vt = Nc (3)

p can be easily computed by substituting (2) into (3) as follows

n∑t=1

mmtkt + pn∑

t=1

(mMt −mmt)kt = Nc (4)

and then solving for p,

p =Nc −

∑nt=1mmtkt∑n

t=1(mMt −mmt)kt(5)

Note that vi takes real values; however, in the case of TDMAsystems, the number of assigned resources (time slots) to theservice group i, wi, is an integer. Hence the actual number ofallocated resources to each group of active calls with servicetype i is wi = f(vi), where f(vi) may be the floor (� · ) orthe ceiling ( · �) function such that (1) is fulfilled. The vec-tor w = {w1, w2, . . . , wn} may have different combinations,though. The vector with the least mean square error relative tov = {v1, v2, . . . , vn} and whose elements meet the conditionkimmi ≤ wi ≤ kimMi (for 1 ≤ i ≤ n) is taken. The reshuf-fling of resources is carried out considering that the incomingcall is accepted.

Determining a fair distribution of resources in the general-ization of ERSA is as simple as determining the value of pgiven by (5), followed by the evaluation of (2). If no value of pcan be found so that all active calls have at least the minimumbandwidth required, irrespective of the service type, then theincoming call is blocked or dropped.

b) Distribution of Resources Between Calls With the SameService Type: Once a set of resources is assigned to each of thendifferent groups of ongoing calls, then resources are distributedas evenly as possible among calls with the same type of service(that is, within each group) in very much the same manner asexplained in [14]. Basically, the number of resources allocatedto group i, (1 ≤ i ≤ n), wi, are as equally distributed as possibleamong active calls with this service type. Hence, a number ofusers wi − ki�wi/ki are allocated �wi/ki + 1 resources, andki(�wi/ki + 1) − wi users are allocated �wi/ki resources. Inthis way, the total number of assigned resources to all activecalls with service type i is:(wi − ki

⌊wi

ki

⌋)(⌊wi

ki

⌋+ 1

)

+(ki

(⌊wi

ki

⌋+ 1

)− wi

)⌊wi

ki

⌋= wi (6)

With this approach, the maximum difference of resourcesallocated to calls with the same type of service is 1.

2) Call Termination: When a call with service type i ends(naturally or forced) in a given cell, resources are released. Then,

a) If there are active calls (of any service type) with a numberof assigned resources lower than the maximum requested,then the released resources are reallocated to them.

2) Otherwise, no action is required.In the first case a), (2) is used to determine the number of

resources required for each of the n groups. Then, the surplusresources are assigned among the active calls within each group.Preference is given to active calls with the least number ofresources. Random selection of a call to compensate might benecessary if two or more active calls are served with the samelow number of resources. If case b) occurs, then all active callsin the cell operate with the maximum bandwidth requested.

Note that the number of resource reallocations executed uponthe call arrival/departure of a type i call is at most mMi. Thus,even when the number of resources allocated to several ser-vice types may vary, at most mMi active sessions are affected(degraded or compensated).

B. The FRAQoS Strategy

The FRA strategy with differentiated priorities and QoS(FRAQoS) differs from other FRA strategies presented inthe literature not only because FRAQoS is specifically de-signed for systems with multiple services, but because italso prioritizes the QoS of particular service types over therest. This is achieved by introducing the concepts of pri-oritized call degradation and prioritized call compensation.Priorities are assigned to service types rather than specificusers. It is up to the operator to decide which services aregiven what priority.14 Prioritized degradation basically means

14Depending on the QoS requirements for various service requests frommobile users, different priorities may be assigned to various connections. Forexample, real-time services such as voice or streaming video may be assignedhigher priority over non-real-time service; handoff call connections should begiven higher priority over new call connections in order to reduce the forcedtermination probability; mission critical data should be handled with higher


that when congestion occurs, active calls of service typeswith the lowest priorities are degraded first. The opposite oc-curs with prioritized call compensation. In this way, no callswith high (low) priority are degraded (compensated) if callswith low (high) priority can be degraded (compensated) inFRAQoS.

When a call requesting service type i arrives at a cell, if thecell has enough resources available, the call is allocated themaximum number of resources it requires (mMi). On the otherhand, if the total number of resources available in the cell (Na)is less than the maximum number of resources required, mMi,then prioritized degradation of calls in progress is carried outto release resources and accept the incoming call. Degradationof a call in progress exists only when a call operates with moreresources than the minimum required and involves the gradualreduction of the number of resources allocated to it. Prioritizeddegradation takes priorities into account. Ongoing calls of ser-vice types with the lowest priority are degraded first and thosewith the highest priority are degraded last. When it is necessaryto degrade ongoing calls of a priority group, a cyclic processbegins in which calls with the largest number of resources aredegraded first, and calls with the least resources last. In orderto distribute the resources within a group of calls with the sameservice type as equally as possible, calls are degraded one re-source at a time. The degradation process for an engaged callends when the number of resources allocated to it has beenreduced to the minimum required, mmi.

When Na < mMi, prioritized degradation is applied in thefollowing order: first to calls with lower priority (i < j), thento calls with the same priority (i = j), and finally to calls withhigher priority (i > j). (i and j represent the service type, orpriority, of the incoming new or handed over call and the call tobe degraded, respectively). Bear in mind that there are n servicetypes (1, 2, . . . , n) numbered in such a way that the servicetype 1 is the one with the highest priority. The three cases areexplained in detail below.

1) Degradation of Calls With Lower Priority (i < j): WhenNa < mMi, calls with lower priority are degraded with theultimate goal of providing the incoming call withmMi resourcesor releasing the largest number of resources possible. Hence,calls of service type j are degraded until Na = mMi or untilall the calls of service type j have mmj resources allocated. Ifthe first condition is met, the call arrival will be assigned mMi

resources. If the second condition is met instead,Na is updatedand the following case is executed.

2) Degradation of Calls With the Same Priority (i = j): Ifdegradation of calls with lower priority does not provide theincoming call withmMi resources, then degradation of ongoingcalls with the same service type is carried out. The primaryobjective of this new case is providing the incoming call withat least mmi resources and sharing as equally as possible thenumber of available resources for calls of service type i.

priorities than some real-time data such as voice; users who pay more for theirservices should be treated with higher priorities over those who pay less [27].Assigning different priorities to the different service types is also a way to protectimportant traffic during emergencies where traffic load increases drastically [39].

After degrading calls with priority j > i, the total number ofresources that calls of service type i (including the incomingcall) could utilize would be Na +RAi. (RAi is the number ofresources assigned to ongoing calls of service type i.) Underthese circumstances, if there were ki active users of service typei, then the number of resources that a call of service type i coulduse would be �(Na + RAi)/(ki + 1) or �(Na + RAi)/(ki +1) + 1. This allows the available resources to be shared asequally as possible.

Degradation of calls of service type i takes place until there are�(Na + RAi)/(ki + 1) available resources for the incomingcall, or until the ki active users are degraded tommi, whicheveroccurs first. If the first condition is met, the incoming call isassigned �(Na + RAi)/(ki + 1) resources. If only the secondcondition is met, then further degradation of calls with higherpriority is necessary.

3) Degradation of Calls With Higher Priority (i > j): Ifthe updated number of available resources Na after degradingcalls of service types j (i ≤ j) is not enough to provide at leastmmi resources, then degradation of calls with priority j (i > j)takes place with the purpose of assigning the incoming callmmi resources. Hence, calls of service type j are degraded untilNa = mmi or until all the calls of service type j have mmj

resources allocated. If the first instance is met, the call arrivalwill be assigned mmi resources. If the second condition is metinstead, the incoming call is blocked or dropped.

It should be stated, however, that calls are not actually de-graded until it is ensured, by means of calculation, that theincoming call would be accepted with at least its minimumbandwidth requirement. Also, as can be deduced from the strat-egy description, FRAQoS provides users of the same servicetype with the same amount of resources whenever possible,producing a uniformization of their QoS.

Note that a call of service type i cannot have mMi resourcesallocated if a call of service type i− 1 has mm(i−1) resourcesallocated. However, the opposite case may be true. That is, callsof service type i− 1 may be allocated mM(i−1) while somecalls of service type i have mmi allocated resources. It is alsoworth noting that with FRAQoS, the higher the priority for aservice type, the lower the risks of degradation it will have.

As was previously stated, prioritized compensation is alsoused by FRAQoS. Compensation is a process triggered by a calldeparture and is the inverse process of degradation. It consistsof the reallocation of resources released by call departures tocalls which quality requirements may be improved (i.e., calls ofservice type j that operate with less than mMj resources). Pri-oritized compensation also takes into account a call’s priority;calls with the highest priority are compensated first. The com-pensation process for an engaged call ends when the number ofresources allocated to it reaches mMi.

Since FRAQoS dynamically distributes cell resources to callrequests based on the number of calls in a cell, most calls in thecell are allocated the minimum amount of bandwidth requiredas the traffic gets high. The scheme, therefore, accommodatesthe maximum number of requests, as does the group hereafterreferred to as maximum capacity (MC) strategies (including H2,ERSA, and ERSAQoS).


C. Teletraffic Analysis

The following analysis does not consider service time de-pendence on the allocated bandwidth. Hence, calls have an un-encumbered call duration according to a negative exponentialdistribution with parameters µ1, µ2, . . . , µn, for service type 1to n, respectively. This assumption would not be valid in caseswhere services exist whose unencumbered call duration dependson the allocated bandwidth (i.e., interactive service class). How-ever, as will be shown in Sections IV and V, the analysis heredeveloped can be easily extended to consider service time de-pendence on the allocated bandwidth in FRA, with uniform QoSprovisioning as well.

A multidimensional birth and death process is used to modelthe MC strategies (this is based on the assumptions of SectionII and because the MC strategies are coordinate convex accesspolicies that allow for the decomposition of state probabilitiesinto a simple product form [25], [26]). The equivalent offeredtraffic for the service type i (for i = 1, . . . , n) calls per cell isgiven by

ai = (λi + λhi)/(µi + ηi) (7)

The incoming handoff attempt rates per cell (λh1, . . . , λhn)are calculated by the iteration method described in [13]. Letus denote the state of the system as (k1, k2, . . . , kn), wherek1, k2, . . . , kn, represent the number of users of service type1, 2, . . . , n, in the cell under study. The state probabilities forcoordinate convex access policies can be decomposed into asimple product form [25]. Then, the probability that the systemis in the state P (k1, k2, . . . , kn) is given by:

P (k1, . . . , kn)

=

∏nt=1

aktt

kt !∑� N cm m 1

k1=0

∑� N cm m 2

k2=0 · · ·

∑� N cm m n

kn =0

{(k1, k2, . . . , kn) |∑n

t=1 ktmmt ≤ Nc }∏n

t=1a

ktt

kt !

(8)

where � · is the floor function. The denominator of (8), whichis obtained from the normalization equation, is the inverse ofthe probability of being in the state (0, 0, . . . , 0). As the mathe-matical product form implies insensitivity to calls’ holding timedistribution, these results are valid for all distributions with thesame mean holding time [36]. As explained in Section II, thehandoff arrival rates are calculated iteratively. The outgoinghandoff arrival rates of the previous iteration are used as inputsof the current iteration (i.e., fixed point iteration).

The new blocking probability for calls of service typei, Pbi, is given by the summation of all the valid statesfor which the condition Nc − (k1mm1 + k2mm2 + · · · +

knmmn) < mmi is fulfilled. This is: (Please see the equation atthe bottom of page.)

Because the cardinality of the set of allowable states growsroughly as (Nc)n, even relatively modest sized problems ruleout a brute force computation of the blocking probabilities (9)as a function of the steady state probabilities given by (8). Thisproblem is addressed in [46]–[48], through the evaluation ofa one-dimensional recursive formula. In [46]–[48], the “macrostates” technique is used to effectively collapse the multidimen-sional system state representation to a one-dimensional “macro-state” sufficient to analyze all quantities of interest.

Since no handoff prioritization strategy is used, the handofffailure probability for each service type, Phi, is equal to thenew call blocking probability of the service type, so Phi =Pbi.15 When the unencumbered service time is exponentiallydistributed, the forced termination probability, Pfti, is givenby [13]:

Pfti =Phi

µi/ηi + Phi(10)

The equilibrium state probabilities and expressions for thenew call blocking and forced termination probabilities for everyservice type derived for the MC strategies is valid for ERSA,ERSAQoS, FRAQoS, and H2.

In FRA with uniform QoS provisioning, the QoS is stronglytied to the average bandwidth (or number of resources) granted.The larger the average bandwidth allocated, the better the QoSexperienced by the final user (in terms of the subjective QoSfor conversational and streaming services, and in terms of thetransmission delay for interactive services). Since most of theFRA strategies evaluated here (H2, ERSA, FRAQoS) provideuniform QoS among users of the same service type, all usersutilize almost the average bandwidth. Hence, this parameterseems to be adequate to measure the QoS provided by FRAstrategies with uniform QoS provisioning. However, note thatmore elaborated QoS measurements could be used, as thoseproposed in [10] and [49]. In ERSA, the average number ofresources allocated to calls of service type i, Bavg i, is given by:(Please see the equation at the bottom of next page.)

The factor 1/(1 − P (ki = 0)) is used to normalize the aver-age number of resources used by calls of service type i and toconsider only the states where there is at least one ongoing callof service type i. Note that if there is not an ongoing call, theaverage number of resources used does not exist. The probabil-ity that there is not any ongoing call of service type i is givenby (Please see the equation at the bottom of next page.)

15The combination of the strategy described with handoff prioritizationschemes is possible and desirable but is out of the scope of this work.

Pbi =

� N cm m 1

∑k1=0

� N cm m 2

∑k2=0

· · ·� N c

m m n∑

kn =0{(k1, k2, . . . , kn)

∣∣∣∣∣n∑

t=1

ktmmt > Nc −mmi

}P (k1, k2, . . . , kn) (9)


In (11), the first term between brackets considers the statesin which all the users with any service type use the maximumnumber of resources requested and therefore no degradationexists. The second term considers the states in which there is atleast a call with a number of resources lower than the maximumrequested.

On the other hand, in the FRAQoS strategy, the averagenumber of resources used by calls of service type 1, Bavg1,is given by: (Please see the equation at the bottom of thepage.)

The first factor of the right hand member, 1/(1 − P (k1 = 0)),is used to normalize the average number of resources used bycalls of service type 1 and to consider only the states where thereis at least one ongoing call of service type 1. The probability

that there is not any ongoing call of service type 1 is given by(Please see the equation at the bottom of the page.)

The first term of additions in (13) accounts for the states inwhich at least one of all the ki active users with service type isuffers degradation. The second term of additions considers thestates in which there is no degradation. Note the conditions ofthe additions. They consider the fact that, in FRAQoS, no callswith higher (lower) priority are degraded (compensated) if callswith lower (higher) priority can be degraded (compensated).

The average number of resources used by calls of service typei, Bavg i, is given by: (Please see the equation at the bottom ofnext page.)

The first term of additions in (15) takes into account the statesin which all the ki active users with service type i use mmi

Bavg i =

� N cm M 1

∑k1=1

· · ·� N c

m M i∑

ki =1

· · ·� N c

m M n∑

kn =0{(k1, · · · , ki, · · · , kn)

∣∣∣∣∣n∑

t=1

ktmMt ≤ Nc

}mMiP (k1,...,ki ,...,kn )

1−P (ki =0)

+

� N cm m 1

∑k1=0

· · ·� N c

m m i∑

ki =1

· · ·� N c

m m n∑

kn =0

wi

ki

P (k1, . . . , ki, . . . , kn)1 − P (ki = 0){

(k1, . . . , ki, . . . , kn)

∣∣∣∣∣(

n∑t=1

ktmmt ≤ Nc

)∩(

n∑t=1

ktmMt > Nc

)}

(11)

P (ki = 0) =

� N cm m 1

∑k1=0

� N cm m 2

∑k2=0

· · ·� N c

m m n∑

kn =0{(k1, . . . , ki = 0, . . . , kn)

∣∣∣∣∣n∑

t=1

ktmmt ≤ Nc

}P (k1, . . . , ki = 0, . . . , kn) (12)

Bavg1 =1

1 − P (k1 = 0)

·

� N cm m 1

∑k1=0

· · ·� N c

m m i∑

ki =1

· · ·� N c

m m n∑

kn =0

(k1, . . . , ki, . . . , kn)

∣∣∣∣∣(k1mM1 +

n∑t=2

ktmmt > Nc

)

∩(

n∑t=1

ktmmt ≤ Nc

)

[ (Nc −∑n

t=2 ktmmt)k1

·P (k1, . . . , ki, . . . , kn)

]

+

� N cm m 1

∑k1=0

· · ·� N c

m m i∑

ki =1

· · ·� N c

m m n∑

kn =0{(k1, . . . , ki, . . . , kn)

∣∣∣∣∣k1mM1 +n∑

t=2

ktmmt ≤ Nc

}mM1P (k1, . . . , ki, . . . , kn)

(13)

P (k1 = 0) =

� N cm m 1

∑k2=0

· · ·� N c

m m n∑

kn =0{(k1 = 0, k2, . . . , kn)

∣∣∣∣ n∑t=1

ktmmt ≤ Nc

}P (k1 = 0, k2, . . . , kn) (14)


resources. The third term of additions considers the states inwhich there is no degradation; that is, all active users of servicetype i use the maximum bandwidth requested. The second termof additions considers those cases where some or all of theki active calls with service type i use a number of resourcesbetween the minimum and the maximum. Clearly, the averagenumber of resources granted to a call of service type i, Bavg i,is mmi in H2.

Even when several simplifications have been done, the anal-ysis can be used to validate preliminary discrete event computersimulation results before more complex considerations are done(i.e., channel reservation, probability distribution of unencum-bered service time other than negative exponential). Bear inmind that the consideration of coordinate convex access poli-cies has allowed the state probability distribution to be expressedin product form [40].

IV. UNENCUMBERED SERVICE TIME DEPENDENCE ON THE

ALLOCATED BANDWIDTH

In this section, the unencumbered service time dependenceon the allocated bandwidth is addressed. First, FRA strategieswithout degradations/compensations are considered. Then FRAstrategies with uniform QoS provisioning are evaluated. Twoanalytical methods are considered. The first method is semian-alytical; the second method is analytical.

A. Strategies Without Degradations/Compensations

When the service time depends on the allocated bandwidth,it is inversely proportional to the allocated bandwidth or thenumber mi (mmi ≤ mi ≤ mMi) of resources used [15]. Inconversational and streaming class services the service time isindependent of the allocated bandwidth and, therefore, the ser-

vice time of that service class does not change with a variationin the allocated bandwidth. In FRA strategies without degrada-tions/compensations, sessions utilize a constant bandwidth fortheir whole duration. In [15], it is proposed that active calls areallowed to change (degrade or compensate) their service rate ateach handover occurrence. However, as the negative exponen-tial distribution is assumed for the unencumbered call duration,the remaining unencumbered service time of a call handed off isonly a function of the number of resources allocated in the newcell. In this work, it is considered that the optimal or minimumunencumbered duration of a session operating at the maximumbandwidth requested for service type i,Xi, can be modeled bya random variable with negative exponential distribution prob-ability density function (n.e.p.d.f.) with mean 1/µiMax. Mathe-matically, the unencumbered service time of a session that uses,on average, Ui = mi resources is obtained when the randomvariable that models the optimal or minimum service time, Xi,is scaled by the constant mMi/mi, resulting in the randomvariable Yi = (mMi/mi)Xi. Then the conditional probabilitydensity function (pdf) of the unencumbered duration of a ses-sion, Yi, for sessions of service type i that usemi resources, isgiven by

fYi(w|Ui = mi) =

mi

mMifXi

(mi

mMiw

)(16)

Thus, as the pdf of the minimum unencumbered duration ofa session is considered a negative exponential, then the con-ditional pdf of Yi remains exponentially distributed but withmean mMi/(miµiMax). That is,

fYi(w) =

miµiMax

mMie

−m i µi Maxm M i

wu(w) (17)

Bavg i =1

1 − P (ki = 0)

·

� N cm m 1

∑k1=0

. . .

� N cm m i

∑ki =1

· · ·� N c

m m n∑

kn =0{(k1, . . . , ki, . . . , kn)

∣∣∣∣∣(

n∑t=1

ktmmt ≤ Nc

)∩(

i−1∑t=1

ktmMt +n∑

t=i

ktmmt > Nc

)}mmiP (k1, . . . , ki, . . . , kn)

+

� N cm m 1

∑k1=0

· · ·� N c

m m i∑

ki =1

· · ·� N c

m m n∑

kn =0

(k1, . . . , ki, . . . , kn)

∣∣∣∣∣(

i∑t=1

ktmMt +n∑

t=i+1

ktmmt > Nc

)

∩(

i−1∑t=1

ktmMt +n∑

t=i

ktmmt ≤ Nc

)

(Nc −

∑i−1t=1 ktmMt −

∑nt=i+1 ktmmt

)ki

·P (k1, . . . , ki, . . . , kn)

+

� N cm m 1

∑k1=0

· · ·� N c

m m i∑

ki =1

· · ·� N c

m m n∑

kn =0{(k1, . . . , ki, . . . , kn)

∣∣∣∣∣i∑

t=1

ktmMt +n∑

t=i+1

ktmmt ≤ Nc

}mMiP (k1, . . . , ki, . . . , kn)

(15)


where u(w) is the step function defined as:

u(w) ={

1 ; w ≥ 00 ; w < 0

Hence, if the number of utilized resources by a session ofservice type i is, for its whole duration (or remaining servicetime when the negative exponential distribution is assumed forthe unencumbered call duration), equal to mi (where mmi ≤mi ≤ mMi), the effective service rate will be µiMaxmi/mMi.Note that the random variable of the unconditional service timehas a hyperexponential distribution [50].

B. Strategies With Uniform QoS Provisioning:The Semianalytical Approach

In FRA strategies with uniform QoS provisioning, the num-ber of resources used by sessions can be viewed as a randomvariable. The transmission delay for the service i is given by

Di = Yi − Xi (18)

where Yi is a random variable representing the actual servicetime, and Xi is a random variable with pdf fXi

(w) representingthe optimum service time (operating at the maximum band-width requested). Mathematically, the unencumbered servicetime of a session that uses, on average for its duration, a ran-dom number Uavg i of resources (ranging frommmi tomMi) isobtained when the random variable that models the optimal orminimum service time, Xi, is multiplied by the random variablemMi/Uavg i. This results in the random variable

Yi =mMis

Uavg iXi (19)

Then, transmission delay given by (18) can be rewritten as

Di = Yi − Xi =mMi

Uavg iXi − Xi = Xi

(mMi

Uavg i− 1

)(20)

Remember that Uavg i is a random variable that representsthe average number of resources used by a session. Also, notethat the second factor on the right hand of (20) represents thenormalized transmission delay, di; that is:

di =Di

Xi=(mMi

Uavg i− 1

)(21)

In the semianalytical approach, the pdf of the normalizedtransmission delay is obtained by means of discrete event com-puter simulation. Then, the pdf of the transmission delay, Di,is given by the pdf of the product of the normalized transmis-sion delay (di) and the optimum or ideal unencumbered servicetime (Xi). From the simulation results (see Figs. 1 and 2), itcan be observed that the pdf of the normalized transmission de-lay shows a peak at a certain value ci and roughly resembles asymmetric function within an interval of normalized transmis-sion delay values around ci (even function around ci).16 If therandom variable that models such normalized delay is discrete

16Notice that the pdf of the normalized transmission delay is roughly sym-metrical for the most likely values of the normalized transmission delay andwithin an interval of normalized transmission delay values around ci .

Fig. 1. Normalized transmission delay for the ERSA strategy at differentoffered traffic loads.

Fig. 2. Normalized transmission delay for the FRAQoS strategy at differentoffered traffic loads.

and takes values (ci ± dk) (for k = 1, . . . , N ), then the pdf ofthe transmission delay (Di) is given by:

fDi(w)

= P{di = ci − dN}(

1ci − dN

)fXi

(w

ci − dN

)+ · · ·

+ P{di = ci − d1}(

1ci − d1

)fXi

(w

ci − d1

)

+ P{di = ci}(

1ci

)fXi

(w

ci

)

+ P{di = ci + d1}(

1ci + d1

)fXi

(w

ci + d1

)+ · · ·

+ P{di = ci + dN}(

1ci + dN

)fXi

(w

ci + dN

)(22)

where

N∑k=1

[P{di = ci ± dk}] + P{di = ci} = 1 (23)


The factor 1/(ci ± dk) of (22) results from scaling the ran-dom variable Xi, that denotes the ideal unencumbered ser-vice time, by a particular normalized transmission delay di =ci ± dk, which occurs with probability P{di = ci ± dk}, fork = 1, . . . , N .

If the probability mass function (pmf) of the normalized trans-mission delay were perfectly symmetrical, then

P{di = ci − dk} = P{di = ci + dk} (24)

As the pmf of the normalized transmission delay is roughlysymmetrical for a limited range of normalized transmission de-lay values around ci, then only some pairs of terms of (22) canbe approximated by:

P{di = ci − dk}(

1ci − dk

)fXi

(w

ci − dk

)

+ P{di = ci + dk}(

1ci + dk

)fXi

(w

ci + dk

)

∼= 2P{di = ci + dk}(

1ci

)fXi

(w

ci

)(25)

In fact, when ci � dk or equivalently, when dk → 0, the in-equality in the previous equation can be replaced by an equality.That is:

limdkci

→0

[P{di = ci − dk}

(1

ci − dk

)fXi

(w

ci − dk

)

+ P{di = ci + dk}(

1ci + dk

)fXi

(w

ci + dk

)]

= 2P{di = ci}(

1ci

)fXi

(w

ci

)(26)

Then, the term on the right of (25) is a good approximationof the sum of the left hand side of the equation for values of thenormalized transmission delay around ci. The accuracy of theapproximation of (25) is numerically illustrated in Appendix.Using (25), (22), and the facts that the pmf of the normalizedtransmission delay is roughly symmetrical only for a limitedrange of normalized transmission delay values around ci andthat the most likely values of the normalized transmission delayoccur around ci, the pdf of the transmission delay (Di), givenby (22), can be approximated by:

fDi(w) ∼= 2P{di = ci − dN}

(1ci

)fXi

(w

ci

)+ · · ·

+ 2P{di = ci − d1}(

1ci

)fXi

(w

ci

)

+ P{di = ci}(

1ci

)fXi

(w

ci

)

=1cifXi

(w

ci

)(27)

As shown below, as the normalized transmission delay haslow variance (i.e., the values of the normalized transmissiondelay around ci are more likely to occur), the pdf of the trans-mission delay (Di) can be accurately approximated by (27).This is simply the pdf of a random variable that models the opti-

TABLE IMEAN AND VARIANCE OF THE NORMALIZED TRANSMISSION DELAY FOR THE

ERSA STRATEGY AT DIFFERENT OFFERED TRAFFIC LOADS

TABLE IIMEAN AND VARIANCE OF THE NORMALIZED TRANSMISSION DELAY FOR THE

FRAQOS STRATEGY AT DIFFERENT OFFERED TRAFFIC LOADS

mum service time scaled by a constant ci [51], which is preciselythe basis of the method proposed in Section IV-C. As will beseen, ci is replaced by the factor (mMi/Bavg i − 1). Note thatthe approximation is accurate because the most common val-ues of the normalized transmission delay occur around ci andthe pmf of the normalized transmission delay is approximatelysymmetrical around ci. Tables I and II show numerical valuesof the mean and variance of the normalized transmission delayfor the ERSA and FRAQoS strategies (obtained from simula-tion results shown in Figs. 1 and 2), respectively, at differentoffered traffic loads. These tables show also the standard devi-ation to mean ratio of the normalized transmission delay (thatis, the standard deviation in units of the mean value). Note thatthe standard deviation is small relative to the mean, and that itdecreases as the offered traffic increases.17

Assuming that Di is a continuous random variable, its pdffDi

(w) can be obtained by doing N → ∞ in (22)

fDi(w)

= limN →∞∆d→0

N∑k=0

[P

{ci − dk − ∆d

2< di ≤ ci − dk +

∆d2

}

·(

1ci − dk

)fXi

(w

ci − dk

)

+ P{ci + dk − ∆d

2< di ≤ ci + dk +

∆d2

}

·(

1ci + dk

)fXi

(w

ci + dk

)]

=∫ Linf

ci

fdi(s)fXi

(w

ci − s

)ds

+∫ Lsup

ci

fdi(s)fXi

(w

ci + s

)ds (28)

For the numerical results, the pdf of di obtained by simulationwas parameterized to compute (28).

17This is due to the facts that the most likely values of the normalized trans-mission delay are around ci (where the pdf of the normalized transmission delaypeaks) and that the domain of the pdf of the normalized transmission delay islimited in the range [0, 1].


C. Strategies With Uniform QoS Provisioning:The Analytical Approximation

In FRA strategies with uniform QoS provisioning, even whenthe sessions of service type i do not use exactly the averagenumber of resources Bavg i (mmi ≤ Bavg i ≤ mMi), all thesessions use a number of resources close to the average num-ber of resources (i.e., the ratio of the total average number ofresources available for a service type to the average numberof active calls of that service type).18 This is due to the rela-tive high resource reassignment rate. Thus, the service time foreach session of service type i is stretched out by a magnitudeclose to the constant mMi/Bavg i, relative to the optimal trans-mission rate (operating at the maximum bandwidth requested).Mathematically, the random variable that models the optimal orminimum service time, Xi, is scaled approximately by the con-stantmMi/Bavg i. Then, the resulting service time random vari-able Yi ≈ Xi(mMi/Bavg i) is obtained where Yi has a similarprobability distribution to Xi. Since Xi is assumed exponen-tially distributed, the pdf of the service time random variableYi is almost exponentially distributed as well, but with a meanmMi/(Bavg iµiMax). Notice that, strictly speaking, the trans-mission time Yi has a hyperexponential distribution [50], whereeach of the negative exponential components of the weightedsum has a similar mean. In fact, in the limit, the hyperexponen-tial distribution tends to an exponential distribution (when allthe negative exponential components have the same mean).

As such, in FRA strategies with uniform QoS provisioning,the transmission delay for service type i sessions, Di, can beapproximated by subtracting the optimal or minimum servicetime, Xi, from the transmission time, Yi; that is,

Di = Yi − Xi∼= mMi

Bavg iXi − Xi = Xi

(mMi

Bavg i− 1

)(29)

Then, the pdf of Di is approximated by [51]:

fDi(w) ∼= 1

mM i

Bavg i− 1

fXi

(w

mM i

Bavg i− 1

)(30)

Thus, the random variable Di has similar pdf to the randomvariable Xi but with mean:

E{Di} ∼= 1µiMax

(mMi

Bavg i− 1

)(31)

as Di is a random variable approximately equal to the scaledrandom variable Xi [51].

V. THE PROPOSED ANALYSIS

This section proposes an approach that reduces the evaluationcomplexity of FRA strategies with uniform QoS provisioningwhere the session service time depends on the allocated band-width. The evaluation is reduced to an assessment of the averagebandwidth used by sessions of each service type, Bavg i. This is

18The system considered in this work is ergodic, hence the average numberof allocated resources to users with service type iBavg i (sample average) canbe used instead of the average number of resources used by the sessions withthe service type i (time average, E{Uavg i}).

done through an iterative process that uses the analysis devel-oped for FRA strategies where the service time does not dependon the number of resources used [expressions derived in SectionIII-C, (7) to (15)], therefore having the same simplicity. How-ever, in order to use the set of (7) to (15) to calculate the averagenumber of resources, the mean unencumbered service times hasto be adjusted to consider the effect of the dependence of theservice time on the allocated bandwidth. This is done by meansof an iterative process in which the values of the average num-bers of resources utilized are guessed. The process ends whenthe calculated average numbers of resources utilized equals theguessed values. The proposed algorithm is as follows.

Inputs: Nc,mmi,mMi, µiMax, ηi, λi, and fi (for i =1, . . . , n).

Outputs: Pb1, Pb2, Pft1, Pft2, Bavg1, Bavg2.

Step 0: Make B(old)avg i = (mMi +mmi)/2 for each service

type i which unencumbered session duration de-pends on the allocated bandwidth, and ε = 0.00001.Go to Step 1.

Step 1: Calculate the steady state probabilities con-sidering the mean service times 1/µi =(µiMaxB

(old)avg i /mMi)−1 if the unencumbered

session duration of service type i depends on theallocated bandwidth or 1/µi = 1/µiMax if theunencumbered session duration of service typei does not depend on the allocated bandwidth(for i = 1, . . . , n). Calculate the average numberof resources used by calls of each service typei (Bavg i) users and make B(new)

avg i = Bavg i (fori = 1, . . . , n). In this step, the analysis developedfor the strategies where the service time call doesnot depend on the number of resources used isutilized, (7)–(15).19 Go to Step 2.

Step 2: The process ends if |B(new)avg i −B(old)

avg i | < εB(new)avg i

for each service type i which unencumbered ses-sion duration depends on the allocated bandwidth.Otherwise, go to Step 3.

Step 3: MakeB(old)avg i = B

(new)avg i for each service type iwhich

unencumbered session duration depends on the al-located bandwidth. Go to Step 1.

The algorithm consists of three iterative loops. The outermost loop determines the values of Bavg i for each service typei which unencumbered session duration depends on the allo-cated bandwidth. The next loop calculates the handoff arrivalrate as described in [13], and the innermost loop calculates thesteady state probabilities using the Gauss-Seidel method, whichare performed within the Step 1. In each iteration stage in theouter most loop, a value of Bavg i is tested for each service

19The expressions (7)–(15) derived in Section III-C for cellular systems withservices whose service time does not depend on the allocated bandwidth arealso applicable for cellular systems with services whose service time dependon the allocated bandwidth. However, the mean unencumbered service timesof the different service calls must be properly adjusted to consider the effectof the dependence of the service time depend on the bandwidth used. This isdone iteratively by guessing values of the average numbers of resources usedand it is stopped until the calculated average numbers of resources used equalthe guessed value.


type i which unencumbered session duration depends on theallocated bandwidth. This is performed until the tested valuefor each Bavg i is equal to the calculated one. If after one itera-tion is executed the calculated value of Bavg i is greater than itstested value, then a larger value has to be tested. Actually, thetested value is updated by assigning it the value just calculated(Step 3). This obeys to the following reason: In order to increasethe calculated value of Bavg i it is necessary to reduce the of-fered traffic, and with a fixed new call arrival intensity λni, thisis achieved by reducing the service time. Service time reductionis achieved by increasing the tested value of Bavg i, as the userscan transmit with a higher bit rate. This procedure incorporatesthe fact that the average number of resources used is a monoton-ically decreasing function of the offered traffic load. Similarly ifthe calculated value ofBavg i is smaller than its tested value afteran iteration; then a smaller value has to be tested. The processconverges when the condition |B(new)

avg i −B(old)avg i | < εB

(new)avg i for

each service type i which unencumbered session duration de-pends on the allocated bandwidth is met.

VI. RESULTS

This section presents numerical results. The evaluations as-sume that there are two service types, and that the unencumberedservice time is negative exponentially distributed. The call ser-vice time does not depend on the allocated bandwidth for servicetype 1 (i.e., voice) but does for service type 2 (i.e., data or fax). Itis also assumed that Nc = 40,mM2 = 2,mm2 = 1,mM1 = 2,and mm1 = 1. In addition, 1/η1 = 900 s, 1/η2 = 1000 s,1/µ1 = 180 s, and 1/µ2Max = 300 s. The proportion of traf-fic of service type 1 is f1 = 0.50.

Four strategies are then evaluated: H2, S1 [16], FRAQoS,and ERSA. Two cases are considered for H2. H2 TD and H2NTD are evaluated to show the upper and lower capacity bounds,respectively, of the performance of FRA strategies with uniformQoS provisioning when considering service time dependence onallocated bandwidth. When H2 NTD is evaluated, it is assumedthat for both service types, the service time does not depend onthe allocated bandwidth (in this case, service type 2 belongs tothe conversational or streaming class and 1/µ2 = 300 s). Onthe contrary, when H2 TD is evaluated, it is assumed that onlythe service time of service type 2 calls depends on the allocatedbandwidth (in this case, service type 2 belongs to the interactiveclass).

Fig. 3 shows Pb1 and Pb2 versus the total offered traffic percell. ERSA TD, FRAQoS TD, and S1 TD denote that servicetime of service type 2 depends on the allocated bandwidth. Ob-serve that both service types have the same new call blockingprobability. This is due to the fact that they have the same block-ing conditions. Figs. 4–5 show Pft1 and Pft2 versus the totaloffered traffic per cell. Remember that H2, ERSA, and FRAQoSprovide the same capacity, and differ only in the policies utilizedto share the resources among the different service types. Notethat the fluctuation of the interactive flows (plots labeled TD)does not result in any load reduction. This is because the amountof data transmitted through an interactive class connection is in-dependent of the available bandwidth (lower bandwidth results

Fig. 3. New call blocking probabilities for strategies where the unencumberedservice times depend on the allocated bandwidth (TD), and for strategies wherethe unencumbered service times do not depend on the allocated bandwidth(NTD) versus the offered traffic per cell.

Fig. 4. Service type 1 forced termination probability for strategies where theunencumbered service times depend on the allocated bandwidth (TD), and forstrategies where the unencumbered service times do not depend on the allocatedbandwidth (NTD) versus the offered traffic per cell.

in longer service time). On the other hand, the fluctuation of theconversational or streaming flows (plots labeled NTD) results ina load reduction, since the amount of data transmitted through aconversational or streaming class connection is proportional tothe available bandwidth during the connection.

Fig. 6 shows the average number of resources used by servicetype 2 users (Bavg2) versus the total offered traffic load per cell.Two cases are considered for the FRAQoS strategy: FRAQoS-1 prioritizes service type 1, while FRAQoS-2 prioritizes ser-vice type 2. Remember that FRAQoS prioritizes one servicetype over the other(s). For the ERSA strategy, because of theequal resource sharing among the different service types and thesame bandwidth requirements,Bavg1 = Bavg2. Note that at low(high) offered traffic loads the different strategies tend to assignthe maximum (minimum) bandwidth requested and, therefore,unencumbered service time of all calls tends toward the optimal(maximum) one. FRAQoS-1 (FRAQoS-2) is the strategy that as-signs the highest average number of resources to users with ser-vice type 1 (2), reaching, at the same time, the MC performance.


Fig. 5. Service type 2 forced termination probability for strategies where theunencumbered service times depend on the allocated bandwidth (TD), and forstrategies where the unencumbered service times do not depend on the allocatedbandwidth (NTD) versus the offered traffic per cell.

Fig. 6. Average number of resources used by service type 2 users versus theoffered traffic per cell.

For an offered load of 24 Erlangs/cell, FRAQoS-2 (FRAQoS-1) produces an increase in Bavg2 (Bavg1) of about 28% withrespect to ERSA, providing better QoS to the service type it pri-oritizes. In FRAQoS, an increase in Bavg1(2) is achieved at theexpense of decreasing Bavg2(1), that is, by degrading anotherservice type(s). With the Maximum Capacity (MC) strategies,the increase of the number of assigned resources to one servicetype is obtained at the expense of resource reduction for otherservice types.

Figs. 7 and 8 show the CDF and pdf of the transmission delayfor service type 2 with the total offered traffic load as a param-eter in the ERSA strategy. The pdf obtained with the proposedanalytical approach given by (30), and with the semianalyticalmethod given by (28), are in agreement with the simulationresults. Notice that as the unencumbered service time is as-sumed to be exponentially distributed, the transmission delay iswell approximated by an exponential distribution. In fact, with

Fig. 7. Cumulative Distribution Function (CDF) for the transmission delay,total offered load as parameter.

Fig. 8. Probability density function (pdf) of the transmission delay.

a 99% confidence interval, the Kolmogorov-Smirnov test wasaccepted. Clearly the analytical approximation is preferred be-cause is simple and in agreement with the simulation results.

VII. CONCLUSION

This paper proposes a novel mathematical analysis approachto evaluate the call level performance of adaptive resource al-location strategies with uniform QoS in multiservice wirelesscellular networks. The paper addresses, in particular, the trans-mission delay in adaptive resource allocation strategies. Trans-mission delay is one of the most important performance mea-surements of interactive and background service classes. Theanalytical approach presented was based on the fact that in FRAstrategies with uniform QoS provisioning, users of a given ser-vice type tend to utilize a bandwidth close to the average amountof resources used by ongoing calls of that service type. The factthat the probability density function (pdf) of the normalizedtransmission delay is almost a symmetrical function, and has


low variance, was also considered. The accuracy of our mathe-matical approach was corroborated by semianalytical results.

The following was observed from the analysis:a) Bandwidth adaptation can achieve better resource utiliza-

tion and improve user service satisfaction (in terms of theallocated bandwidth).

b) The fluctuation of the interactive flows does not result inany load reduction because the amount of data transmittedthrough an interactive class connection is independent ofthe available bandwidth.

c) In contrast, the fluctuation of the conversational or stream-ing flows does result in a load reduction, since the amountof data transmitted through a conversational or streamingclass connection is proportional to the available bandwidthduring the connection.

The results of this analysis are expected to be applicableto more sophisticated system models. Possible extensions ofthis work include addressing dynamic change in the channelconditions [52], packet switched traffic, and data buffering. Allthese issues, along with others considerations (such as the casewhen the new channel holding time and the handoff call channelholding time have different distributions or different averagevalues), are subjects of further research.

APPENDIX

NUMERIC EXAMPLES OF (25)

With the aim of showing the validity of the approximation of(25), some numerical results are shown in this appendix. If themass probability function (mpf) of the normalized transmissiondelay is perfectly symmetric with respect to ci, then (25) can bewritten as follows:(

1ci − dk

)fXi

(w

ci − dk

)

+(

1ci + dk

)fXi

(w

ci + dk

)∼=(

2ci

)fXi

(w

ci

)(32)

If we assume that the ideal unencumbered service time forcalls with service type i has a negative exponential distributiongiven by:

fXi(w) = µe−µwu(w) (33)

Then, substituting (33) in (32) we obtain,(µ

ci − dk

)e−µ( w

ci −dk)u

(w

ci − dk

)

+(

µ

ci + dk

)e−µ( w

ci +dk)u

(w

ci + dk

)

∼=(

2µci

)e−µ( w

ci)u

(w

ci

)(34)

Equation (34) is then evaluated for two different scenarioswhere the mean service time is assumed to be 1/µ = 180 s. Thefirst scenario considers ci = 0.5 and different values of dk. Thesecond scenario considers ci = 0.8 and different values of dk.

Figs. 9 and 10 show results for the scenario 1 and 2, respec-tively. In the scenario 1 (scenario 2): dk = 0.4, dk = 0.3, dk =

Fig. 9. Evaluation of (25) for scenario 1 using different values of dk .

Fig. 10. Evaluation of (25) for scenario 2 using different values of dk .

0.2, anddk = 0.1 (dk = 0.2, dk = 0.16, anddk = 0.12). In bothFigs. 9 and 10, it can be observed that for the cases whereci increases so that ci � dk (or equivalently, when the ratiodk/ci → 0), (34) tends to be an equality rather than an approx-imation. In other words, the difference between the exact valueand the approximation increases. However, since the probabil-ity mass function (pmf) of the normalized transmission delayis concentrated around ci, the probability that dk takes a largevalue is insignificant and, as such, its values have little influencein the transmission delay probability density function (pdf).

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir valuable comments and suggestions, which enhanced thequality of the paper.


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Lauro Ortigoza-Guerrero received the B.Sc. degreein electronics and communications engineering fromESIME-UPC, in 1993, and the M.Sc. degree in elec-trical engineering from Center for Research and Ad-vanced Studies (CINVESTAV), in 1996, both fromInstituto Politecnico Nacional (IPN), Mexico City,Mexico. He received the Ph.D. degree in electric en-gineering from King’s College, London, Universityof London, London, England, in 1999.

In 1998, he joined King’s College London as aResearch Assistant. Since 2000, he has been with

Wireless Facilities, Inc. in San Diego, CA, where he has been a Technical Con-sultant to multiple companies. He has coauthored three books on mobile radiocommunications, and his areas of interest include mobile cellular communica-tion networks with emphasis in radio resource management, teletraffic analysis,and system performance evaluation.

Felipe A. Cruz-Perez (S’98–M’02) was born inMixquiahuala, Hidalgo, Mexico, in 1972. He re-ceived the B.Sc. degree from the Technological In-stitute and Superior Studies of Monterrey (ITESM),Mexico, in 1994, and the M.Sc. and Ph.D. degreesfrom CINVESTAV-IPN, Mexico City, Mexico in1997 and 2001, respectively, all in electrical engi-neering.

Currently, he is with the CINVESTAV-IPN, andhis research interests are in resource management,teletraffic analysis, quality of service provisioning,

call admission control, and prioritized resource allocation in mobile wirelesscommunication systems, microcellular systems, CDMA cellular systems, andwireless communication systems with link adaptation.

Heraclio Heredia-Ureta (S’03) was born in Cu-liacan, Sinaloa, Mexico, in 1977. He received theB.Sc. degree in electronics and communications en-gineering from Instituto Tecnologico de Culiacan,Culiacan, Mexico, in 2000, and the M.Sc. degree inelectrical engineering from CINVESTAV-IPN, Mex-ico City, Mexico, in 2003.

Currently he is with Universidad de OccidenteCampus Culiacan, and his research interests are inresource management, teletraffic analysis, and prior-itized resource allocation in mobile wireless commu-

nication systems.

Call Level Performance Analysis for Multiservices Wireless Cellular Networks With Adaptive Resource...

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