Blocking Probability Analysis of Circuit-Switched Networks With Long- Lived and Short-Lived...

Blocking Probability Analysis ofCircuit-Switched Networks With Long-Lived and Short-Lived Connections

Meiqian Wang, Shuo Li, Eric W. M. Wong, and Moshe Zukerman

Abstract—We consider a circuit-switched network withnonhierarchical alternate routing and trunk reservationinvolving two types of connections that are modeled aslong-lived and short-lived calls. The long-lived calls canbe reserved well in advance, and the short-lived callsare provided on demand. Therefore, we assume that thelong-lived calls have strict priority over the short-livedones. We develop approximations for the estimation ofthe blocking probability based on the quasi-stationaryapproach in two ways. One uses the Erlang fixed-pointapproximation (EFPA), and the other uses the overflow pri-ority classification approximation (OPCA). We compare theresults of the approximations with simulation results anddiscuss the accuracy of the approximations under differentsystem parameters, such as ratio of offered load, number oflinks per trunk, maximumallowable number of deflections,and trunk reservation.We also discuss the robustness of thequasi-stationary approximation to the ratio of the meanholding times of the long-lived and short-lived calls as wellas that of EFPA and OPCA to the shape of the holding timedistribution. Finally, we demonstrate that OPCA can beapplied to a large network such as the Coronet.

Index Terms—Alternate routing; Blocking probability;Circuit switching; Erlang fixed-point approximation;Optical flow switching; Overflow priority classificationapproximation; Quasi-stationary approximation; Trunkreservation.

I. INTRODUCTION

C ircuit switching has been widely used in telephony,and it is envisaged that it will have a renewed and

important role in future optical networks [1–5]. Circuitswitching normally does not require buffering, which isvery costly in the optical domain. If the traffic on a circuit-switched network is sufficiently heavy and it is well man-aged, such a network can guarantee quality of service (QoS)to customers in a way that can lead to efficient trunkutilization and low consumption of energy per bit [6]. Inthe core Internet, where traffic is heavily multiplexed, itis easier to achieve high utilization, and therefore the roleof circuit switching at the core is clearly important. Circuitswitching can also lead to a green and efficient operation

end-to-end for large bursts of data if the amount of datato be transmitted is known in advance.

Circuit-switched networks may encounter traffic over-load situations when the offered traffic exceeds the net-work capacity, and in such situations, there is a need toreject (block or drop) new calls. The proportion of blockedtraffic is defined as blocking probability. Such blocking(or dropping) has adverse implication on the QoS perceivedby users, and therefore blocking probability should notexceed a certain predetermined value. For over a century,operators have considered blocking probability as a keyperformance measure for circuit-switched network designand dimensioning. In today’s competitive environment,limiting blocking probability is clearly important.

To reduce blocking probability in circuit-switched net-works, various approaches for dynamic or alternate routinghave been studied [7–9], where new calls that cannot beadmitted by their primary routes may overflow to otherroutes that may be more costly in terms of the use of net-work resources. The use of more and more costly alternateroutesmay adversely affect blocking probability. Therefore,the number of allowable overflow attempts by a particularcall may be limited, so a call is blocked after a finite numberof alternate route attempts.

While traffic flows are transmitted optically on the dataplane, the configuration of an optical cross connect (OXC) isdone by an electronic control unit. The control units of dif-ferent OXCs can also communicate with each other andcompose the control plane, which is responsible for light-path setup and teardown, as well as routing, label distri-bution, and state dissemination using protocols, such asGMPLS and RSVP-TE [10].

A circuit-switched network with alternate routing,under the assumptions of Poisson call arrivals for anysource destination (SD) pair and exponential call holdingtimes, can be modeled as a Markovian overflow loss net-work. The transition rates between states of a trunk areobtained considering arrival and departure rates of connec-tions on their primary path and those that overflow to thetrunk from other fully occupied trunks. The stationaryoccupancy distribution can, in principle, be obtained by anumerical solution of the steady-state equations of a multi-dimensional Markov process. Such models usually do notadmit product-form solutions [11] and are not amenableto analysis that leads to a scalable solution for realistic sizehttp://dx.doi.org/10.1364/JOCN.5.000621

Manuscript received November 8, 2012; revised April 20, 2013; acceptedApril 26, 2013; published May 31, 2013 (Doc. ID 179527).

The authors are with the Department of Electronic Engineering, CityUniversity of Hong Kong, Hong Kong SAR, China (e-mail: [email protected]).

Wang et al. VOL. 5, NO. 6/JUNE 2013/J. OPT. COMMUN. NETW. 621

1943-0620/13/060621-20$15.00/0 © 2013 Optical Society of America

networks. Therefore accurate, robust, and scalable approx-imations are important.

Circuit-switched networks with alternate routing can beclassified into two classes: hierarchical and nonhierarchi-cal. In hierarchical networks, the trunks are ranked intoseveral tiers. New calls first attempt to access trunks fromthe lowest tier; if rejected, they overflow and attempt to ac-cess trunks from higher tiers. For hierarchical networks,accurate blocking probability approximations can beobtained using moment matching approaches [12,13].

A more difficult problem is to accurately approximateblocking probabilities in nonhierarchical networks withmutual overflows [7], where congestion on a specific trunkcauses overflow to the other trunks, and where this over-flow loads up the other trunks so that they in turn yieldoverflow back to the original trunk. Clearly, in such non-hierarchical systems, load dependencies may be muchstronger than in hierarchical systems that do not havemutual overflow, and modeling involving independenceassumptions is more likely to lead to significant errors.Despite wide applicability and importance, and a century-long research effort, no robust and generic methodology isavailable for approximation of the blocking probability ofgeneral nonhierarchical networks that captures theiroverflow-induced state dependencies in a scalable way, ex-cept for studies for particular applications [14–16].

One simple and commonly used approach for approxima-tion of blocking probabilities in nonhierarchical networksis the Erlang fixed-point approximation (EFPA) [9,17–20],proposed by Cooper and Katz [21] in 1964. It is based ondecoupling a given system into independent subsystems,each of which is modeled by anM/M/K/K queueing system.Convergence and uniqueness of solutions of EFPA is notalways guaranteed [9,22]. Despite its popularity, EFPAis known to introduce two types of errors, Poisson errorand independence error [23]. Various attempts to addressthe errors include [21,24], which provided means to reducethe Poisson error by moment matching, and [25], whichtackled the independence error by capturing the correla-tion between trunks.

In recent years, we have developed a new method calledoverflow priority classification approximation (OPCA)[23,26–29] that can either improve or complement EFPAto achieve more accurate blocking probability approxima-tions for circuit-switched networks with alternativerouting.

In this paper, we retain the assumption that holdingtimes are independent and follow identical exponentialdistribution as in most of the published work on analysisof circuit-switched networks. However, we assume thatthe calls are classified into two types, long-lived (static)and short-lived (dynamic), where the holding times ofthe long-lived calls are significantly larger than those ofthe short-lived calls. This assumption is justified by the factthat the variations in holding times in circuit-switched op-tical networks can be significant. Permanent or semiper-manent connections between major cities or data centers[1], which are used to serve many flows of many users overa long period of time, can be considered as (long-lived) calls

where the holding time can be in terms of hours. Examplesof such long-lived connections can also include circuit-switched connections in the large hadron collider (LHC)network [2], which can be modeled as long-lived calls.On the other hand, on-demand dynamic connections be-tween individual SD pairs of users in the order of secondsor less may be classified as short-lived calls. Classificationof calls to long-lived and short-lived classes have beendiscussed in various papers (e.g., [30–33]).

Long-lived connections are likely to have priority overthe short-lived ones, as long-lived connections can bebooked well in advance, so it is reasonable to assume thattheir performance will not be affected by the loading of theshort-lived ones. Furthermore, long-lived connections be-tween major cities or data centers carry traffic from manyusers, so it is justifiable for them to have preemptive prior-ity over the short-lived ones. Otherwise, the long-livedconnections have to compete with the much more frequentshort-lived demands in the same pool of resources, so theirblocking probability will increase significantly, which willbe detrimental to the many customers they serve. Such pri-ority is also likely to be given to connections that serve thelarge data bursts generated by the LHC [2], which were thekey reason for the design of the LHC network. Then lowerpriority short-lived connections can share the remainingcapacity.

These assumptions justify the use of the so-called quasi-stationary approximation [34–36] that are suitable in thecases where changes in system states observed by one typeof traffic, due to changes in other traffic type(s), are rare.

To the best of our knowledge, the problem of blockingprobability evaluation of circuit-switched networks withalternate routing and multiple priorities assigned toshort-lived and long-lived calls has not been studied before.In this paper we apply both OPCA and EFPA to evaluatethe blocking probability of the two traffic types in a circuit-switched network with alternate routing. We compare theresults obtained against simulation benchmarks and ex-plain their performance in various different scenariosand parameter ranges. Then we discuss the insight gainedinto performance trade-offs as well as design and dimen-sioning implications.

The remainder of the paper is organized as follows. InSection II, we provide a detailed description of our networkmodel and define notation and basic concepts. Next, inSection III, we provide algorithms based on EFPA andOPCA to evaluate the blocking probability of the model.Then, in Section IV, we provide numerical results over awide range of parameters for two network topologies [fullymeshed and National Science Foundation (NSF) network]and discuss performance and design implications. Finally,the paper is concluded in Section V.

II. MODEL

We consider a circuit-switched network described by agraph G�N;E�, where N is a set of n nodes and E is theset of e arcs. The e arcs correspond to trunks where trunk

622 J. OPT. COMMUN. NETW./VOL. 5, NO. 6/JUNE 2013 Wang et al.

i ∈ E carries C�i� links. The N nodes are designated1;2; 3;…; N, and each of them has circuit switching capa-bilities. We assume that all the nodes have full wavelengthconversion capabilities and can switch traffic from any linkon one trunk to any other link on an adjacent trunk.

In the context of a core WDM network, a wavelengthchannel can be viewed as a link. In this case, trunk i ∈E is composed of f �i� fibers, each of which supports w�i�wavelengths. Accordingly, trunk i ∈ E carries C�i� �f �i�w�i� wavelength channels called links. However, ifthe WDM network is further extended to the metropolitanor local areas, a link can have a subwavelength capacity[37]. The assumption above that all nodes can switch trafficfrom any link on one trunk to any other link on anothertrunk, in the WDM context, implies that we assume thatall switches have full wavelength conversion capabilities.In principle, our model can be extended to exclude thisassumption, as we can, in our model, split every switchand trunk to multiple “subswitches” and “subtrunks,” eachof which is dedicated to one color wavelength. Then we canincrease the number of allowable alternate routes by a fac-tor of the number of wavelengths. However, this implies asignificant increase in the computational complexity of oursolutions as the graph that describes the network and thenumber of alternate routes significantly increase. Katiband Medhi [38] studied by simulations the trade-off be-tween alternate routing and the number of convertersfor single priority networks. The focus of this paper is ondeveloping accurate approximations for blocking probabil-ity for networks with alternate routing and long-lived andshort-lived calls.

Let Γ be a set of directional SD pairs. Every directionalSD pair m ∈ Γ is defined by its end nodes. Thus, m �fs; dg ∈ Γ represents the directional SD pair s to d. We willdistinguish between the term “SD pair,” which is an unor-dered set of the two endpoints, source and destination, andthe “directional SD pair” that refers to the ordered set,source–destination.

The calls are classified according to their priority p(p � 1, 2). Long-lived calls (p � 1) have preemptive priorityover short-lived calls (p � 2). For each directional SD pairm ∈ Γ, calls of priority p arrive according to a Poissonprocess with arrival rate λ�m;p� [39]. The holding timesof calls are assumed exponentially distributed with mean1∕μ�m;p� [40]. We assume that holding times are exponen-tially distributed for tractability. However, it is well knownthat in loss systems, blocking probability is highly insensi-tive to the shape of the holding time distribution and onlydependent on themean value of the holding times. This hasbeen proven for the M/G/K/K system. In Section IV, wehave also demonstrated numerically that the blockingprobability of our model is also insensitive to the shapeof the holding times in Subsection IV.A. Let

ρ�m;p� � λ�m;p�μ�m;p�

be the offered traffic (measured in Erlangs) for directionalSD pair m. We set

ρ�p� �Xm∈Γ

ρ�m;p�:

A route between source s and destination d is the sequenceof trunks associated with the corresponding arcs in thepath between s and d in G�N;E�. A path between s andd comprises a sequence of trunks—one on each trunk onthe route between s and d.

It is very likely that for a directional SD pairm ∈ Γ, thereare multiple routes between the source and the destinationthat do not share a common trunk. Such routes are oftencalled edge-disjoint paths or disjoint paths [41–43]. Edge-disjoint alternate routing is often used to achieve loadbalancing in optical and other networks [44,45]. Althoughusing disjoint paths has the benefit of reducing blockingprobability, nondisjoint paths are often used in practice.Our approximation methods are also applicable to thecases where paths are not disjoint. However, at this stagethe strong dependency between trunks in this case, whichmay increase network blocking probability, causes ourapproximation methods to underestimate the blockingprobability. Development of algorithms to improve accu-racy in the case of nondisjoint paths is still an open prob-lem. In this paper, we only consider disjoint paths in thenumerical examples presented in Section IV.

For eachm ∈ Γ, we designate a route with the least num-ber of hops as the primary path U�m; 0� of the directionalSD pair m. If there are multiple routes with the least num-ber of hops, for tractability, the choice is made randomlywith equal probabilities. Also, in practice, it may havethe advantage of keeping the routing table unchanged.Then considering a new topology where the trunks ofthe primary path are excluded, the first alternative pathfor m is chosen to minimize the number of hops in thenew topology. Again, a tie is broken randomly. Therefore,all the paths form, including one primary path and severalalternative paths, are edge disjoint. Let Tm be the maxi-mum number of available alternative paths a directionalSD pair m can have based on the network topology.

Furthermore, a maximal number D of overflow attemptsto alternate paths are set for calls from all directional SDpairs in Γ. Setting the limit D implies that a call of thedirectional SD pair m can only use

T�m� � minfTm;Dg

alternative paths. Therefore, before a call is blocked, theprocedure continues until all available and allowableT�m� routes are attempted.

It is convenient to maintain the entire set

fU�m; 0�; U�m; 1�;…; U�m;T�m��g

of alternative routes for the directional SD pair m ∈ Γ inwhich U�m; 0� is the primary path and U�m;d� is thedth alternate path. This allows for cases where D doesnot limit the number of usable alternative paths.

In our model, the ranking of alternative paths is basedon the number of hops, and in the case of equality in thenumber of hops, the rank is chosen randomly. Based on


our ranking, if d�u� > d�v�, then the number of hops ofU�m;d�u�� is equal to or higher than the number of hopsin U�m;d�v��. However, in practice, other cost functions(e.g., geographic distance) can also be used for the ranking.

If a request for a call arrives at source node s to the des-tination node d, and capacity is available on all trunks ofthe primary path U�fs; dg; 0�, then this primary path willbe used for the transmission of this call.

An arriving call of any type can use any free link on anytrunk. When a long-lived call arrives, it can obtain a pathon the primary path if no trunk of its primary path has allthe links used by long-lived calls. In this case, if it finds allthe links in any trunk on its primary path busy, it will pre-empt a randomly chosen short-lived call. Then the pre-empted short-lived call will release its resource to thelong-lived call and overflow to its next alternate path. If thearriving high priority long-lived call finds that all the linksare occupied by higher priority calls on at least one of thetrunks of the primary path, it will attempt a route on thefirst alternate path. In such a case, the long-lived call issaid to be overflowed from its primary path and to attemptits first alternate path. The same procedure is repeated un-til the long-lived call exhausts all its D alternate path at-tempts. Then if it still cannot obtain a path, the call will beblocked and cleared of the network.

When a short-lived call arrives, it can obtain a path onthe primary path if no trunk of its primary path has all thelinks used by either long-lived or short-lived calls. Other-wise, it will overflow to its first alternate path. Again,the same procedure is repeated. If it is not able to obtaina path in its D alternate path attempts, the call is blockedand cleared of the network.

We realize that call reattempts can affect blocking prob-ability. However, normally our intention is to dimension thenetwork so that the blocking probability is maintained be-low a certain small value. In this case, the proportion of callreattempts out of the total arriving calls is small, and theireffect on the blocking probability is negligible.

Considering stability of the network, and recognizingthat fewer resources are used by a call that uses its primarypath, priority is given to such calls. To facilitate such prior-ity, a certain number of unoccupied links are reserved forcalls attempting their primary path. In particular, if thenumber of links occupied on trunk j is greater than or equalto a given reservation threshold RT� j; p�, the overflowedcalls of priority p are not allowed to use that trunk. Inthe paper, we use trunk reservation to reserve a certainnumber of links to primary path calls to avoid a large num-ber of overflowed calls in the network, which may cause in-stability. In all the numerical examples that are presentedin the paper (excluding the case of the large Coronet net-work discussed in Subsection IV.K), both EFPA and OPCAhave converged to a unique solution.

III. BLOCKING PROBABILITY APPROXIMATIONS

In this section we describe the approximations we usefor blocking probability evaluation for the long-lived and

short-lived calls. We use the term “0 call” for a call trans-mitted on its primary path, and the term “d call” for acall transmitted on its dth alternate path, ford � 1; 2;…; T�m�. Accordingly, the term “�d;m; p� call” re-fers to a d call of priority p, p � 1, 2, and directional SDpair m, in which for long-lived traffic p � 1 and forshort-lived traffic p � 2. Assume that the arrivals of the�d;m; p� calls follow a Poisson process with ratea�d;m; p� and the arrivals of the �d;m; p� calls at trunkj ∈ U�m;d� also follow a Poisson process with ratea�d;m; p; j�. And if j is the first trunk on the path of the�d;m; p� calls, then a�d;m; p; j� � a�d;m; p�. Let b�d; j; p�be the blocking probability for priority p d calls ontrunk j ∈ E.

The �d;m; p� calls occur only when �d − 1;m; p� calls areblocked. Therefore, we have

a�d;m; p� � a�d − 1;m; p��1 −

Yj∈U�m;d−1�

�1 − b�d − 1; j; p��

(1)

and a�0;m; p� � ρ�m;p�. For a particular trunk along thepath j ∈ U�m;d�, we have

a�d;m; p; j� � a�d;m; p�Q

i∈U�m;d��1 − b�d; i; p��1 − b�d; j; p� (2)

for d � 0; 1;…; T�m�. For d > T�m� or j ∉ U�m;d�,a�d;m; p; j� � 0.

Let a�d; j; p� be the total offered load of priority p d calls,on trunk j. They are related to a�d;m; p; j� by

a�d; j; p� �Xm∈Γ

a�d;m; p; j�: (3)

Also, let ~a�d; j; p� be the total offered load of priority pcalls that include 0 calls, 1 calls, 2 calls, …, d calls, ontrunk j. The variables ~a�d; j; p� and a�d; j; p� are related by

~a�d; j; p� �Xdi�0

a�i; j; p�: (4)

We provide key symbol notation in Table I.

TABLE ISUMMARY OF NOTATION

d call A call transmitted on its dth alternate path�d;m; p�call

d call of priority p (p � 1, 2) and directionalSD pair m

ρ�m;p� Offered load of SD pair m and priority pa�d;m; p� Offered load of �d;m; p� callsa�d;m; p; j� Offered load of �d;m; p� calls on trunk ja�d; j; p� Total offered load of priority p d calls on trunk j~a�d; j; p� Total offered load of priority p up to d calls

on trunk jb�d; j; p� Blocking probability for priority p d calls

on trunk jB�m;p� Blocking probabilities for priority p traffic from

SD pair mB�p� Network blocking probability for priority p traffic


Since long-lived traffic has preemptive priority overshort-lived traffic, the blocking probability of higher prior-ity long-lived traffic can be evaluated as if it were alone inthe network. In other words, it is sufficient to consider anetwork with a single class of traffic and apply EFPAand OPCA directly to it for the estimation of blocking prob-ability of long-lived traffic. For lower priority short-livedtraffic, the available capacity is the leftover of long-livedcarried traffic, and therefore the blocking probability ofshort-lived calls in a trunk is dependent on the numberof links occupied by long-lived calls in the trunk. To evalu-ate the blocking probability of the short-lived traffic, we usea quasi-stationary approximation in both EFPA and OPCAand calculate the conditional blocking probability of short-lived traffic for each state of long-lived traffic occupancyand then compute the weighted average of these probabil-ities using the stationary distribution of the long-lived traf-fic link occupancy.

A. EFPA

To obtain the estimation of the network blocking proba-bility for the long-lived traffic by EFPA, we begin by settinginitial randomly chosen Uniform[0,1) values to trunkblocking probabilities, and the probability that a call is ad-mitted on a trunk is 1 minus the trunk blocking probability.With these initial values, we treat the trunks as if theywere independent and the probability that a call is admit-ted to the primary path is the product of probabilities that acall is admitted to all the individual trunks along the path.The primary path blocking probability is 1 minus the prob-ability that a call is admitted to the path. The carried traf-fic on the primary path for a particular SD pair is theunblocked proportion of the offered traffic, which is alsothe carried traffic on each trunk along the path attributedto this SD pair. Accordingly, the offered load of the SDpair to a trunk is obtained by the carried traffic attributedto that SD pair divided by 1 minus the trunk blockingprobability.

The traffic overflowed to the first alternative path is thetraffic offered to the primary path, multiplied by the pri-mary path blocking probability and, in the same way, traf-fic offered to other alternative paths can also be obtained.Having obtained the traffic offered to every SD pair thatuses the trunk, the total traffic offered to the trunk is theirsum. Then, we calculate the steady-state probabilities ofeach trunk in the network and update the trunk blockingprobabilities with the ones obtained based on the stateprobabilities. All these steps compose one iteration ofthe fixed-point equations. The iterations will continueuntil the difference of the trunk blocking probabilities ofthe successive two iterations is less than a given value.Having obtained the trunk blocking probabilities, we canobtain network blocking probability for the long-livedtraffic.

Let q� j; 1; i� be the steady-state probability of having ilinks busy by long-lived traffic in trunk j. We evaluatethe trunk state probability q� j; 1; i�, for each j and i ∈f1;…; C� j�g by

q� j; 1; i� ��a�0; j; 1� � 1fRT� j; 1� > i − 1g

XDn�1

a�n; j; 1��

× q� j; 1; i − 1�∕i; (5)

where 1f g is the indicator function and q� j; 1;0� is set suchthat

PC� j�i�0 q� j; 1; i� � 1 is satisfied. The blocking probabil-

ity, for the long-lived traffic with d overflows, on trunk j isestimated by

b�d; j; 1� ��q� j; 1; C� j�� d � 0;PC� j�

i�RT� j;1� q� j;1; i� d ≥ 1: (6)

To evaluate the blocking probability for the lower prior-ity short-lived traffic, we use the quasi-stationary approxi-mation (in the sense of [34–36,46]). Such an approximationis often used when the variations in system state observedby one type of traffic are very rare. In our case, as the hold-ing times of long-lived calls are far longer than those of theshort-lived calls, so that short-lived calls only rarelyobserve changes in their service rate during their holdingtime and can approximately reach steady-state while thenumber of long-lived calls remains unchanged. Under suchconditions, accurate approximation for the blocking proba-bility for the short-lived calls can be obtained by computingthe short-lived traffic blocking probability for each stateof the long-lived traffic trunk occupancy and then comput-ing the weighted average of these probabilities using thestationary distribution of the long-lived traffic trunk occu-pancy. A more detailed description of this approximationfor our case follows.

To obtain the network blocking probability approxima-tion for the short-lived traffic byEFPA, theprocedure is sim-ilar to that for the long-lived traffic. The difference is thesteady-state probabilities and the blocking probabilitiesof the short-lived traffic are conditional on the numberof links that are occupied by the long-lived calls onthat trunk.

Let b�d; j; k; 2� be the blocking probabilities for the short-lived calls with d overflows for each trunk j when there arek ∈ f0;…; C� j�g links free in trunk j.

First, set: b�d; j; 0;2� � 1, for each d, and also ford � 1; 2;…; D, when k ≤ C� j� − RT� j; 2�, set b�d; j; k; 2� � 1.

Next, to evaluate other b�d; j; k; p� values, we evaluatethe trunk state probability q� j; k; 2; i� for each trunk jand each state i ∈ f1;…; kg using

q� j; k; 2; i� ��a�0; j; 2� � 1fR� j; k� > i − 1g

XDn�1

a�n; j; 2��

× q� j; k; 2; i − 1�∕i; (7)

where R� j; k� � RT� j; 2� � k − C� j� and q� j; k; 2;0� is setsuch that

Pki�0 q� j; k; 2; i� � 1 is satisfied.

Then we obtain

b�d; j;k;2� �8<:1 k� 0 or R� j;k�≤ 0;q� j;k;2;k� d� 0 and R� j;k�> 0;Pk

i�R� j;k� q� j;k;2; i� d≥ 1 and R� j;k�> 0:

(8)


The blocking probability, for the short-lived traffic with doverflows, on trunk j is estimated by

b�d; j; 2� �XC� j�i�0

q� j; 1; i� × b�d; j; C� j� − i; 2�: (9)

Equations (1)–(9) form a set of fixed-point equations thatcan be solved by successive substitutions.

Having obtained the results of the fixed-point equations,we calculate the blocking probabilities for the long-livedtraffic (p � 1) and short-lived traffic (p � 2) from SD pairm by

B�m;p� � 1 −XDd�0

a�d;m; j; p��1 − b�d; j; p��∕ρ�m;p�; (10)

where j is the last trunk in the route for the calls of SDpairmthat overflow d times. Let B�p� be the network blockingprobability for long-lived traffic (p � 1) and short-livedtraffic (p � 2), which is the average of blocking probabilitiesof all SD pairs, weighted by their offered load:

B�p� �Xm∈Γ

B�m;p� × ρ�m;p�.X

m∈Γρ�m;p�: (11)

The relative error is a parameter, set to measure the dif-ference of the substitution results, and the iteration willstop when

Xj∈E

jb�d; j; 1� − b̂�d; j; 1�j < error: (12)

In this paper, we set

error � 10−8: (13)

Algorithm 2 describes the computation of the networkblocking probability for the short-lived traffic, which isdependent on the state probability q� j; 1; i� computed byAlgorithm 1.

Algorithm 1 Compute B�1� and q� j;1;i� by EFPARequire: ρ�m; 1� for m ∈ Γ

initial: b�d; j; 1�←0, b̂�d; j; 1�←1 for j ∈ E, d ∈ f0;…; Dgwhile

Pd∈f0;…;Dg

Pj∈E jb�d; j; 1� − b̂�d; j; 1�j > error do

for j ∈ E, d ∈ f0;…; Dg, m ∈ Γ, dob̂�d; j; 1�←b�d; j; 1�compute a�d;m;1� in Eq. (1)compute a�d;m; j; 1� in Eq. (2)compute a�d; j; 1� in Eq. (3)for i ∈ f1;…; C� j�g docompute q� j; 1; i� in Eq. (5)end forcompute b�d; j; 1� in Eq. (6)end forend whilefor m ∈ Γ docompute B�m; 1� in Eq. (13)end forcompute B�1� in Eq. (11)

Algorithm 2 Compute B�2� by EFPARequire: ρ�m; 1�, ρ�m; 2� for m ∈ Γ, q� j; 1; i� for j ∈ Eand i ∈ f1;…; C� j�ginitial: b�d; j; 2�←0, b̂�d; j; 2�←1 for j ∈ E, d ∈ f0;…; Dgwhile

Pd∈f0;…;Dg

Pj∈E jb�d; j; 2� − b̂�d; j; 2�j > error do

for j ∈ E, d ∈ f0;…; Dg, m ∈ Γ dob̂�d; j; 2�←b�d; j; 2�compute a�d;m; 2� in Eq. (1)compute a�d;m; j; 2� in Eq. (2)compute a�d; j; 2� in Eq. (3)for k ∈ f1;…; C� j�g, i ∈ f1;…; kg docompute q� j; k; 2; i� in Eq. (7)compute b�d; j; k; 2� in Eq. (8)end forcompute b�d; j; 2� in Eq. (9)end for

end whilefor m ∈ Γ docompute B�m; 2� in Eq. (13)

end forcompute B�2� in Eq. (11)

EFPA is known to introduce two types of errors:

1) The Poisson error—EFPA assumes that the trafficoffered to any trunk follows a Poisson process, whereas,in fact, the traffic offered by an overflow call is known tohave a higher variance than a Poisson process [13],when traffic offered to a sequence of trunks on a pathmay be smoothed out when offered to one trunk dueto blocking in another trunk.

2) The independence error—EFPA assumes thattrunks are mutually independent, whereas they are,in fact, statistically dependent.

Algorithm 1 is used to obtain the network blocking prob-ability B�1� for the long-lived traffic.

As both the Poisson and the independence errors are rela-ted to two effects that are characteristics of circuit-switchednetworks, namely, an effect associated with overflows andthe effect associatedwith the fact that a call requires amulti-hop path to be established. We will henceforth call errors ofEFPA causedby these two effects overflow errorsand path er-rors. While overflow errors cause underestimation of block-ing probability (ignoring high variance of overflow trafficand dependence), path error overestimates blocking proba-bility because it ignores the effect of traffic smoothing, andthepositivecorrelationoftrunkoccupancyalongthepaththatincreases the probability to admit calls. These relationshipsbetween the errors and their effects are shown in Table II.

B. OPCA

OPCAworksbyusingahierarchical surrogatesecondsys-tem and estimating the blocking probability in thesecond system by an EFPA-like algorithm. The surrogatesystem is defined by regarding an overflow loss networkas if it were operating under a preemptive priority regimewhereeach call is classifiedaccording to thenumberof timesit has overflowed and junior calls (calls that experienced


fewer overflows) are given priority. By giving priority tojunior calls, the senior calls that have more “information”about busy paths, namely, paths where at least one trunkis busy (all the links in that trunk are busy), are preemptedand overflowed, and these senior calls will only attemptalternate paths that they did not visit before. In this way,the surrogate system operates as a hierarchical networkwhere the traffic is strictly prioritizedand layeredaccordingto how many times it overflows. We remind the readerthat the prioritization introduced in the surrogate systemis artificially introduced to obtain a more accurate approxi-mation, and it is not a feature of the real network.

Despite the fact that the surrogate system may be differ-ent from the real system we aim to analyze, the applicationof EFPA to the surrogate system can, in many cases, pro-vide a better blocking probability approximation for theoriginal problem than the application of EFPA to the origi-nal problem due to the following reasons:

1) OPCA avoids the adverse effects on accuracy of mutualoverflow.

2) OPCA increases the proportion of the 0 calls in the sys-tem and reduces the overflowed traffic. Since 0 calls donot violate the Poisson and independence assumptions,increasing the proportion can reduce the Poisson andindependence errors.

3) Having more primary traffic reduces also the independ-ence errors.

The procedure of the blocking probability calculation byOPCA is similar to that ofEFPA, and the difference betweenthe two is due to the preemptive priority of the surrogatemodel of OPCA. In OPCA, we first solve the fixed-pointequations considering only traffic offered to the primarypath that has not overflowed yet, andwe obtain the networkblocking probability for it. Then, we calculate the total traf-fic that comprises the traffic that has not overflowed and thetraffic that has overflowed once. We again solve the fixed-point equations and obtain the blocked portion of the totaltraffic. Subtracting the traffic that was blocked once in theprimary path from the total blocked traffic, we obtain thetraffic that is blocked twice, which gives the blocking prob-ability for the traffic that has overflowed once. Blockingprobabilities of traffic that has overflowed more than onceare obtained recursively in a similar way.

In the following we provide detailed information on howto apply OPCA to the present problem of approximatingthe blocking probability of circuit-switched networks forthe long-lived and short-lived traffic.

We begin by evaluating the trunk state probabilityt�d; j; p; i� of long-lived traffic (p � 1) for each trunk j, ford deflections and each state i ∈ f1;…; C� j�g using

t�d; j; 1; i� ��a�0; j; 1� � 1fRT� j; 1� > i − 1g

Xdn�1

a�n; j; 1��

× t�d; j; 1; i − 1�∕i; (14)

where t�d; j; 1;0� is set to satisfyPC� j�

i�0 t�d; j; 1; i� � 1.

The average blocking probability b̄�d; j;1� on trunk j, forthe long-lived calls with up to and including d overflows, isestimated by

b̄�d; j; 1� �Pd

n�1�a�n; j; 1�PC� j�

i�RT� j;1� t�d; j; 1; i��~a�d; j; 1�

� a�0; j; 1�t�d; j; 1; C� j��~a�d; j; 1� (15)

and b̄�0; j; 1� is estimated using the Erlang-B formula, i.e.,b̄�0; j; 1� � E�a�0; j; 1�; C� j��. The blocking probability forthe long-lived traffic, for d overflow calls, on trunk j is esti-mated by

b�d; j; 1� �(b̄�0; j; 1� d � 0;b̄�d;j;1� ~a�d;j;1�−b̄�d−1;j;1� ~a�d−1;j;1�

a�d;j;1� 1 ≤ d ≤ D:(16)

Note that the blocking probability for the unoverflowedcalls is calculated using the Erlang-B formula. Having ob-tained the trunk blocking probability b�d; j; 1�, the networkblocking probability can be computed by Eqs. (13) and (11).Algorithm 3 is used to compute the network blocking prob-ability for the long-lived traffic.

After calculating all the blocking probabilities fordifferent layers for the long-lived traffic, we move on tocalculate the blocking probabilities for the short-lived traffic.

Algorithm 3 Compute B�1� and h� j;i� by OPCARequire: ρ�m; 1� for m ∈ Γfor d ∈ f0;…; Dg doinitial: b�d; j; 1�←0, b̂�d; j; 1�←1 for j ∈ Ewhile

Pj∈Ejb�d; j;1� − b̂�d; j; 1�j > error do

for j ∈ E, m ∈ Γ dob̂�d; j; 1�←b�d; j; 1�compute a�d;m;1� in Eq. (1)compute a�d;m; j; 1� in Eq. (2)compute a�d; j; 1� in Eq. (3)compute ~a�d; j; 1� in Eq. (4)for i ∈ f1;…; C� j�g docompute t�d; j; 1; i� in Eq. (14)end forcompute b̄�d; j; 1� in Eq. (15)compute b�d; j; 1� in Eq. (16)end forend while

end forfor m ∈ Γ do

compute B�m; 1� in Eq. (13)end forcompute B�1� in Eq. (11)for j ∈ E, i ∈ f1;…; C� j�g do

compute h� j; i� in Eq. (17)end for

TABLE IIEFPA ERRORS

Overflow Error Path Error

Poisson error Underestimate OverestimateIndependence error Underestimate Overestimate


Let h� j; i� for i ∈ f1;…; C� j�g and each trunk j be theprobability that there are i number of long-lived calls intrunk j. So that

h� j; i� ��a�0; j; 1� � 1fRT� j; 1� > i − 1g

XDn�1

a�n; j; 1��

× h� j; i − 1�∕i; (17)

where h� j; 0� is set such thatPC� j�

i�0 h� j; i� � 1 issatisfied.

Let b̄�d; j; k; 2� be the blocking probabilities for the short-lived calls with d overflows for each trunk j when there arek ∈ f0;…; C� j�g links free in the trunk j. For d � 0;1;…; D,b̄�d; j; 0;2� � 1. For d � 1;2;…; D, when R� j; k� �k − C� j� � RT� j; 2� ≤ 0, b̄�d; j; k; 2� � 1. To evaluate otherb̄�d; j; k; p�, we evaluate the trunk state probabilityt� j; k; d;2; i� for each trunk j, each state i ∈ f0;1;…; kg,and d using

t�d; j; k; 2; i� ��a�0; j; 2� � 1fR� j; k� > i − 1g

Xdn�1

a�n; j; 2��

× t�d; j; k; 2; i − 1�∕i; (18)

where t�d; j; k; 2; 0� is set such thatPk

i�0 t�d; j; k; 2; i� � 1 issatisfied, and t�0; j; k; 2; k� is estimated using the Erlang-Bformula, i.e., t�0; j; k; 2; k� � E�a�0; j; 2�; k�.

Then we obtain

b̄�d; j; k;2� �8<:t�0; j; k;2; k� d� 0;Pk

i�R� j;k� t�d; j; i;2; i� d ≥ 1 and R� j; k�> 0;1 otherwise:

(19)

The averaged blocking probability, for the short-lived traffic with d overflows, on trunk j is estimatedby

b̄�d; j; 2� �XC� j�i�0

h� j; i� × b̄�d; j; C� j� − i;2�: (20)

The blocking probability for the short-lived d calls ontrunk j is estimated by

b�d; j; 2� �(b̄�0; j; 2� d � 0;b̄�d;j;2� ~a�d;j;2�−b̄�d−1;j;2� ~a�d−1;j;2�

a�d;j;2� 1 ≤ d ≤ D:(21)

Then the network blocking probability can be computedby Eqs. (13) and (11). Algorithm 4 is used to compute thenetwork blocking probability B�2� for the short-livedtraffic.

Algorithm 4 Compute B�2� by OPCARequire: ρ�m; 1�, ρ�m; 2� for m ∈ Γ, h� j; i� for j ∈ Eand i ∈ f1;…; C� j�gfor d ∈ f0;…; Dg doinitial: b�d; j; 2�←0, b̂�d; j; 2�←1 for j ∈ Ewhile

Pj∈Ejb�d; j;2� − b̂�d; j; 2�j > error do

for j ∈ E, m ∈ Γ dob̂�d; j; 2�←b�d; j; 2�compute a�d;m;2� in Eq. (1)compute a�d;m; j; 2� in Eq. (2)compute a�d; j; 2� in Eq. (3)compute ~a�d; j; 2� in Eq. (4)for k ∈ f1;…; C� j�g, i ∈ f1;…; kg docompute t�d; j; k; 2; i� in Eq. (18)compute b̄�d; j; k; 2� in Eq. (19)end forcompute b̄�d; j; 2� in Eq. (20)compute b�d; j; 2� in Eq. (21)end forend while

end forfor m ∈ Γ do

compute B�m; 2� in Eq. (13)end forcompute B�2� in Eq. (11)

IV. NUMERICAL RESULTS

We begin this section by comparing the performance ofOPCA to the performance of EFPA in approximating theblocking probability for the long-lived and short-livedtraffic using simulations. The comparison is performedfor a 6-node fully meshed network and the 13-node NSFnetwork.

Then, we extend our comparison, to consider a range ofscenarios and parameter values for each scenario. In allcases considered, we also provide an intuitive explanationfor the discrepancies between the two approximations andsimulation results for the blocking probability as it variesaccording to the various effects. In particular, we considertraffic effects such as offered load, and the difference be-tween the two types of traffic, in terms of the offered loadand mean holding times. The latter is especially importantto observe the accuracy and sensitivity of quasi-stationaryapproximation.

Finally, we consider design factors, such as the numberof links per trunk, the maximal allowable number of alter-nate paths, and the effect of trunk reservation. We also dis-cuss the robustness of the approximations to the shape ofthe holding time distribution.

In all scenarios considered, the arrival process of calls foreach directional SD pair and each type of traffic follows aPoisson process and the total traffic offered to each direc-tional SD pair is equal to T. The variable T is our measureof traffic load for all cases in both topologies. The shortestpath is set to be the primary route for each SD pair, and thealternate routes are preassigned, ordered by their length.


For those routes with the same lengths, the order is chosenrandomly and remains unchanged afterward.

A. Default Parameter Setting

In many experiments we repeatedly use the same set ofparameters with possibly small variations. It is convenientto present them once in this section and through the sectiononly point out the deviations from this default set. This setof parameters will henceforth be referred to as the “defaultparameter setting” and is described as follows.

For both the 6-node fully meshed network and the NSFnetwork, both long-lived and short-lived calls arrive ac-cording to a Poisson process, and the ratio of offeredlong-lived traffic to that of short-lived traffic is 1∶1. Theholding times of both long-lived and short-lived calls areexponentially distributed, and the mean holding time oflong-lived traffic is 200 times higher than that of theshort-lived traffic. The total number of links per trunk is20. The threshold of long-lived traffic is 16 (80%), andthe threshold of the short-lived traffic is 18 (90%). Themaximal allowable number of alternate paths is set to 4for both long-lived and short-lived calls in the 6-node fullymeshed network and to 2 in the NSF network, respectively.

We have chosen to present the results, for each network,for the long-lived and short-lived calls in two separatefigures because they are different by several orders of mag-nitude, and if presented on the same figure as a function oftraffic load, for reasonable traffic loads that give acceptableblocking probability to short-lived traffic, in many cases,the blocking probability for the long-lived traffic will betoo low for accurate evaluation by simulation.

B. Blocking Probability for the Long-Lived Traffic

We evaluate here the blocking probability for the long-lived traffic by EFPA, OPCA, and simulations. Sincelong-lived traffic has preemptive priority over short-livedtraffic, the blocking probability for the long-lived trafficcan be evaluated as if it were alone in the network. In otherwords, it is sufficient to consider a network with a single

class of traffic. We note that the case of a single class oftraffic was considered in [27]. Here we add examples, intui-tive explanations, and interpretation of the results.

At first, we consider a 6-node fully meshed networkmodel. In such a network, there are in total 15 differentSD pairs (or equivalently 30 directional SD pairs). As theoffered traffic for each directional SD pair is T, so the totaloffered traffic in the network is 30T.

In Fig. 1(a), we present results for the blocking probabil-ities obtained by OPCA, EFPA, and simulations for long-lived traffic (single priority). We observe in the figure thatEFPA and OPCA tend to underestimate the blocking prob-ability when the offered load is low. This is due to the factthat in a fully meshed network with low traffic load, andtherefore fewer overflows, long paths will be very rare.Accordingly, overflow error will dominate path errors caus-ing underestimation of the blocking probability.

Since the surrogatemodel of OPCA gives preemptive pri-ority to new calls, and therefore allocates more resources tothe primary path traffic, this surrogatemodel has less over-flow traffic than the original model approximated by EFPA,leading to less overflow error, less Poisson error, and there-fore less underestimation of blocking probability for thelong-lived traffic (single priority). Recall that OPCA isbased on approximating the blocking probability of thesurrogate model, treating each link independently andassuming it is loaded by Poisson arrivals.

Furthermore, we observe that as the traffic load in-creases, the underestimation for both EFPA and OPCAof the blocking probability is reduced. This is consistentwith the fact that in high load overflow probability in-creases, leading to overflow path length growth, and there-fore the path error increases. As observed, the path error inthe cases of high traffic load may cancel out the overflowerror to improve the approximation.

Next, we consider an NSF network with 13 nodes and 16bidirectional trunks. As the number of SD pairs is�13 × 12∕2� � 78, the total offered traffic in this case is156T. The topology of the NSF network is shown in Fig. 2.The results for the blocking probabilities for the long-livedtraffic as a function of T is shown in Fig. 1(b). We obtainresults that have similar behavior to those obtained for

Fig. 1. Blocking probabilities for the long-lived traffic (single priority) versus T in (a) a 6-node fully meshed network and (b) an NSFnetwork.


the 6-node fully meshed network case in the sense thatOPCA outperforms EFPA. But since the number of alter-nate paths in the NSF network is much less than thatin the 6-node fully meshed network, in the NSF network,OPCA outperforms EFPA less than in the case of the6-node fully meshed network.

C. Blocking Probability for the Short-Lived Traffic

Figures 3(a) and 3(b) present the blocking probabilities’results we obtained by OPCA, EFPA, and simulations forshort-lived traffic, as a function of T, in the same 6-nodefully meshed and NSF networks described above, butnow we consider the two classes of traffic using the net-works so T is now the total offered traffic of both long-livedand short-lived calls for each directional SD pair. Theparameters are set as in the default parameter settingin Subsection IV.A.

As expected, comparing Figs. 3(a) and 3(b) with theirlong-lived traffic (single priority) counterparts Figs. 1(a)and 1(b), respectively, we observe significantly lower block-ing probability (by several orders of magnitude) for thelong-lived traffic than for short-lived traffic. Specifically,observe the blocking probability in Fig. 3(a) for a load ofT � 12 versus the equivalent load of T � 6 in Fig. 1(a).Also, for the NSF network, we compare the blocking prob-ability in Fig. 3(b) for a load of T � 0.5 to that of Fig. 1(b) fora load of T � 0.25.

Furthermore, we observe that OPCA provides better ap-proximations for both long-lived traffic and short-livedtraffic in both networks than EFPA.

If the threshold of long-lived traffic is larger than orequal to that of short-lived traffic, since long-lived callscan preempt short-lived calls and not vice versa, the setof states in which long-lived calls are blocked is a strict sub-set of the set of states in which short-lived calls are blocked,and since both processes arrive in accordance with Poissonprocesses, blocking probability for the long-lived trafficmust be lower than that of short-lived traffic even ifthe offered long-lived traffic is much larger than that ofshort-lived traffic. In general, alternate routing furtherbenefits long-lived traffic, because longer alternateroutes of long-lived traffic, which use more network resour-ces per bit than primary path traffic, have a more detri-mental effect on short-lived traffic than on long-livedtraffic.

To provide some protection to short-lived traffic, we setthe threshold of short-lived traffic larger than that of long-lived traffic so that if the offered short-lived traffic is muchsmaller than that of long-lived traffic, short-lived trafficmay have a smaller blocking probability than long-livedtraffic. Notice that this threshold difference does not affectthe “right” of the long-lived traffic to behave as if it is alonein the system according to their own threshold limitationon overflowed calls. This limitation also provides someprotection to short-lived traffic from overflowed long-livedtraffic using a long alternative path inefficiently.

However, in the present case, the offered short-lived traf-fic is the same as that of long-lived traffic; thus the blockingprobability for the short-lived traffic is larger than that forthe long-lived traffic.

The preemptive property of long-lived traffic affects theapproximations of short-lived traffic in two distinct ways:

1) The capacity available to short-lived traffic is the left-over of long-lived carried traffic, and short-lived trafficcan be preempted by long-lived calls. Therefore, short-lived calls may be forced to take longer alternate routes,so the proportion of the overflow traffic in the total

Fig. 2. NSF network topology, where each solid line represents abidirectional trunk between two nodes.

Fig. 3. Blocking probabilities for the short-lived traffic in (a) a 6-node fully meshed network and (b) an NSF network. The ratio of theoffered long-lived traffic to offered short-lived traffic is 1∶1.


short-lived traffic is higher than for long-lived traffic,leading to higher overflow error and path error. Forlow traffic, since the overflow error is dominant, thishigher proportion of overflow traffic in the totalshort-lived traffic will cause further underestimationof the blocking probability for the short-lived trafficfor both EFPA and OPCA. On the other hand, in a highloading scenario, since the path error is dominant inhigh loading, this higher proportion will cause furtheroverestimation of the blocking probability for the short-lived traffic for both EFPA and OPCA.

2) Long-lived carried traffic, which can be viewed as long-lived background traffic for the short-lived traffic,exhibits dependencies among various trunks. More spe-cifically, the congestion of long-lived traffic on one trunkis likely to cause congestion of long-lived traffic on othertrunks. This congestion dependence of long-lived traffic(background traffic) on trunks in turn causes depend-ence in capacity limitation for short-lived traffic amongdifferent links, which leads to congestion dependence ofshort-lived traffic. However, in the quasi-stationary ap-proach, long-lived carried traffic is assumed to be inde-pendent. This introduces another kind of dependenceerror, which does not occur for long-lived traffic, thatcauses a further underestimation of the blocking prob-ability for the short-lived traffic for both OPCA andEFPA. This effect causes further underestimation forlow traffic for both EFPA and OPCA. As the total of-fered load increases, this effect tends to underestimate

the blocking probability obtained by EFPA and OPCA,which will cancel out the overestimation due to the firsteffect and lead to accurate predictions by both EFPAand OPCA.

D. Effect of the Ratio Between the Offered Long-Lived Traffic and Offered Short-Lived Traffic

Figures 3(a) and 4(a)–4(d) show the blocking probabil-ities for the short-lived traffic when the ratios of the offeredlong-lived traffic to the offered short-lived traffic are 1∶1,1∶2, 1∶5, 2∶1, and 5∶1, respectively, while all the otherparameters are kept the same as in the default parametersetting in Subsection IV.A. As long-lived traffic is not af-fected by short-lived traffic, we only consider here the ac-curacy of OPCA and EFPA for short-lived traffic.

Several observations emerge from Figs. 3(a) and4(a)–4(d):

1) OPCA is generally more accurate than EFPA.2) The accuracy of OPCA in predicting blocking probabil-

ity for the short-lived traffic decreases when the propor-tion of long-lived traffic increases.

3) EFPA accuracy is not significantly affected by the ratioof the two traffic types.

4) Both EFPA and OPCA accuracy increases with in-creased traffic load.

Fig. 4. Blocking probabilities for the short-lived traffic in a 6-node fully meshed network. The ratio of the offered long-lived traffic to theoffered short-lived traffic is (a) 1∶2, (b) 1∶5, (c) 2∶1, and (d) 5∶1.


Observations 1 and 4 are consistent with what we haveobserved in the case when the proportion was 1∶1, and theexplanations above are applicable. To explain observations2 and 3, recall that EFPA under a light load suffers fromthe dependency error. This fact is invariant to the propor-tion between the offered long-lived traffic and the offeredshort-lived traffic. However, OPCA is able to reduce thedependency error of short-lived traffic (notice that thequasi-stationary approximation assumes independence be-tween the long-lived and short-lived traffic for both OPCAand EFPA). Therefore, if the short-lived traffic is reduced,the ability of OPCA to neutralize the dependence effects isalso reduced.

Since OPCA has better performance in the cases of lowshort-lived loading and EFPA has better performance inthe cases of high loading, maxfOPCA;EFPAg is the bestapproximation. In [28], we obtained a similar conclusionthat maxfOPCA;EFPAg is the best approximation for anoptical burst switched network.

For the NSF network with 13 nodes and 16 trunks, theresults for the blocking probabilities for the short-livedtraffic are shown in Figs. 3(b) and 5(a)–5(d) when the ratiosof the offered long-lived traffic to the offered short-livedtraffic are 1∶1, 1∶2, 1∶5, 2∶1, and 5∶1, respectively, whileall the other parameters are kept the same as in the defaultparameter setting in Subsection IV.A. We obtain resultsthat have similar behavior to those obtained for the 6-nodefully meshed network case in the sense that OPCA slightlyoutperforms EFPA in the cases of low loading.

E. Effect of the Number of Links on Each Trunk

To examine the effect of the number of links (wavelengthchannels) on each trunk on blocking probability and on theaccuracy of EFPA and OPCA, we increase now the numberof links on each trunk to 50 in the 6-node fully meshed net-work we consider above. In particular, we consider a sce-nario where the thresholds for long-lived traffic andshort-lived traffic are 40 (80%) and 45 (90%), respectively,while all the other parameters are kept the same as in thedefault parameter setting in Subsection IV.A.

In Fig. 6(a) we provide the results obtained for the block-ing probabilities for the long-lived traffic. We can observethat the accuracy of EFPA is improved compared to thecase of 20 links per trunk shown in Fig. 1(a). The improve-ment in accuracy is achieved because of the followingreasons:

1) When the number of links on each trunk increases, thevariance of the overflow traffic decreases, leading to alower Poisson error.

2) The increase of the number of links on each trunk alsoreduces the proportion of overflowed traffic and there-fore reduces the overflow error, which also increases theaccuracy of EFPA.

We also observed that OPCA, in general, is superior toEFPA, so it is sandwiched between EFPA and the simula-tion results.

Fig. 5. Blocking probabilities for the short-lived traffic in an NSF network. The ratio of the offered long-lived traffic to the offeredshort-lived traffic is (a) 1∶2, (b) 1∶5, (c) 2∶1, and (d) 5∶1.


Figure 6(b) shows the blocking probability for the short-lived traffic. We observe again that the accuracy of EFPA isimproved compared to the case of 20 links per trunk shownin Fig. 3(a) and OPCA is generally sandwiched betweenEFPA and the simulation results. The reasons for theimprovement in EFPA results are the same as thosediscussed in the case of the long-lived traffic blockingprobability evaluation.

Notice also that for the blocking probability evaluationfor both long-lived and short-lived traffic, OPCA still out-performs EFPA in the case of 50 links per trunk. This, to-gether with the improved accuracy as the number of linkson each trunk increases from 20 to 50, provides some evi-dence that OPCA can be accurate as the network capacityscales upward and performs even better than for networkswith lower capacity.

We further increase the number of links on each trunk ina 6-node fully meshed network, and in Fig. 7(a), we providethe results obtained by EFPA and OPCA for the blockingprobabilities for the long-lived traffic versus the number oflinks on each trunk. The threshold of long-lived traffic iskept at 80%, and the maximum allowable alternate pathis kept at 4. Let C be the number of links on each trunk.The offered long-lived traffic (Erlangs) is 0.4C for each di-rectional SD pair, so the total traffic per SD pair in bothdirections is 0.8C. As increasing the traffic and the numberof links on each trunk at the same rate will decrease theproportion of overflowed traffic, in large capacity networks,

only negligible traffic is overflowed, so that almost all end-to-end paths in our fully meshed networks are single link(the network approximately turns into a fixed-routingnetwork), and this will mean that approximately all linkswill be 80% utilized. In fixed-routing large capacity net-works, EFPA is accurate based on Kelly [9]. Since thereis only negligible overflow in large capacity networks, asmentioned in the Introduction, we have the conditionunder which the model is based on fixed routing; i.e., nooverflow is allowed, and OPCA is reduced to EFPA. Thisis consistent with the results presented in Fig. 7(a).

Figure 7(b) shows the blocking probability for the short-lived traffic versus the number of links per trunk. Thethresholds of long-lived traffic and short-lived traffic areset as before at 80% and 90%, respectively, while all theother parameters are kept the same as in the defaultparameter setting in Subsection IV.A. The offered trafficload for both long-lived and short-lived calls for each direc-tional SD pair are 0.2C. Although, in general, we observesimilar results for short-lived traffic blocking probability tothose obtained for long-lived traffic, we notice a slight dif-ference between the EFPA and OPCA results for the casesof 50 and 100 links per trunk, which indicates that someoverflows at these levels occur, which lead to conditionswhere EFPA underestimates the blocking probabilityslightly more than OPCA.

We also increase the number of links per trunk for theNSF network to 50. In particular, we consider a scenario

Fig. 6. Blocking probabilities for (a) long-lived traffic (single priority) and (b) short-lived traffic in a 6-node fully meshed network with 50links in each trunk.

Fig. 7. Blocking probabilities for (a) long-lived traffic (single priority) and (b) short-lived traffic in a 6-node fully meshed network versusthe number of links on each trunk.


where the thresholds for long-lived traffic and short-livedtraffic are 40 (80%) and 45 (90%), respectively, while all theother parameters are kept the same as in the defaultparameter setting in Subsection IV.A. In Fig. 8(a) we pro-vide the results obtained for the blocking probabilities forthe long-lived traffic, and the blocking probabilities for theshort-lived traffic is shown in Fig. 8(b). The trends andbehavior of the results presented for the case of an NSFnetwork are consistent with the results provided for the6-node fully meshed network case.

We also increase the number of links on each trunk inthe NSF network, and in Fig. 9(a), we again provide theresults obtained by EFPA and OPCA for the blocking prob-abilities for the long-lived traffic versus the number of linksper trunk. These results are based on having the offeredlong-lived traffic (Erlangs) to be 0.02C for each directionalSD pair; the threshold and the maximum allowable alter-nate paths are kept the same as in the default parametersetting in Subsection IV.A.

Then we consider the NSF network with long-livedand short-lived traffic where the thresholds of long-livedtraffic and short-lived traffic are set as before at 80%and 90%, respectively, while all the other parameters arekept the same as in the default parameter setting inSubsection IV.A. Both the offered long-lived traffic andthe offered short-lived traffic for each directional SD pairare 0.01C. In Fig. 9(b) we present the blocking probabilities

for the short-lived traffic versus the number of links pertrunk obtained by EFPA and OPCA.

Comparing Fig. 9(a) with 7(a) and Fig. 9(b) with 7(b), theresults for an NSF network are generally consistent withthose for the 6-node fully meshed network. The resultsbased on OPCA and EFPA are almost identical. The smalldiscrepancy observed for short-lived traffic in the case of afully meshed network does not exist in the present case be-cause under NSF the allowable number of alternate routesis smaller, so this scenario is closer to a fixed-routing net-work than the fully meshed alternate-routing network, inwhich case we already know that if large capacity is avail-able on trunks, both OPCA and EFPA are very accurate.

F. Effect of Maximal Allowable Number ofAlternate Paths

Here we examine how the blocking probabilities are af-fected by the maximal allowable number of alternatepaths. The maximal allowable number of alternate pathsD limits how many times traffic can overflow. Traffic thatalready overflowedD times is not allowed to overflow againand will be blocked and cleared of the network. For singleclass networks with light traffic, increasing D appropri-ately means more opportunities to overflow and benefitsthe system by reducing the blocking probability. However,when the offered load in the network is high, increasing D

Fig. 8. Blocking probabilities for (a) long-lived traffic (single priority) and (b) short-lived traffic in an NSF network with 50 links in eachtrunk.

Fig. 9. Blocking probabilities for (a) long-lived traffic (single priority) and (b) short-lived traffic in an NSF network versus the number oflinks per trunk.


may not reduce the blocking probability because of the in-efficiency associated with having the average number oflinks used per call unnecessarily long.

Figure 10(a) demonstrates the effect of the maximalallowable number of alternate paths on the blockingprobability for the long-lived traffic obtained by simulation,EFPA, and OPCA. The offered long-lived traffic is 6.3Erlangs, and the threshold is 16 (80%). Figure 10(b) dem-onstrates the effect of the maximal allowable number of al-ternate paths on the blocking probability for the short-livedtraffic obtained by simulation, EFPA, and OPCA. We focuson the 6-node fully meshed network as the low averagenode degree of NSF network restricts the number of alter-nate paths and therefore the number of overflows. The of-fered traffic load for both long-lived and short-lived trafficare 3.6 Erlangs. We change the maximal allowable numberof alternate paths, while keeping all the other parametersthe same as in the default parameter setting inSubsection IV.A.

We observe that there is a clear benefit, in the presentexample, of a 6-node fully meshed network, to increasethe maximal number of overflow to at least 2. After that,the rate of decrease in the blocking probability forboth long-lived and short-lived traffic slows down as Dincreases, due to the inefficiency of the long alter-nate paths.

G. Effect of Trunk Reservation

In general, thresholds are applied to reserve channels forthe primary path traffic and prevent the network frombeing crowded by the overflow traffic. Also, the thresholdof long-lived traffic can protect the short-lived traffic frombeing preempted by long-lived overflow traffic, which re-quires longer paths to establish a call, uses more resources,and may cause congestion. The threshold of long-lived traf-fic should be chosen carefully because a small threshold forlong-lived traffic will cause many overflow calls to be un-able to enter the network, leading to a large blocking prob-ability for the long-lived traffic while the large thresholdinvites too much overflow traffic, congesting the networkand preempting short-lived calls.

We again consider a 6-node fully meshed network, andthe offered long-lived traffic and the offered short-livedcalls are both 3.6 Erlangs. We change the threshold oflong-lived traffic, while keeping all the other parametersthe same as in the default parameter setting inSubsection IV.A.

For this case, Fig. 11(a) illustrates the effect of thethreshold for long-lived traffic on the blocking probabilityfor the short-lived traffic.

We can observe that when both the offered long-livedtraffic and the offered short-lived traffic are equal, varyingthe threshold of the long-lived traffic does not affect theblocking probability for the short-lived traffic. This is be-cause the blocking probability for the long-lived traffic isvery small, and therefore the change of the carried loadof long-lived traffic (background traffic of short-lived traf-fic) caused by the change of threshold for long-lived trafficis also very small, so its effect on blocking the probabilityfor the short-lived traffic is negligible.

Figure 11(b) shows the effect of changing the threshold ofthe long-lived traffic on the blocking probability for theshort-lived traffic when the ratio of the offered long-lived traffic and the offered short-lived traffic is 5∶1. Theoffered loads are set to be 6 and 1.2, for long-lived andshort-lived traffic, respectively, and all the other parame-ters are the same as in the default parameter setting inSubsection IV.A. We can observe that when the long-livedtraffic is much higher than the short-lived traffic, changingthe threshold of the long-lived traffic will increase theblocking probability for the short-lived traffic, due to thefact that the blocking probability for the long-lived trafficis in a range that increasing the threshold of long-livedtraffic will considerably reduce the blocking probabilityand increase the carried load of long-lived traffic (whichis the background traffic of short-lived traffic).

H. Robustness of the Quasi-StationaryApproximation

As stated in Section III, when the holding times of long-lived calls are far longer than those of the short-lived calls,

Fig. 10. Blocking probability for (a) long-lived traffic (single priority) and (b) short-lived traffic for a 6-node fully meshed network.


we can use the quasi-stationary approximation to obtainaccurate results. We use a 6-node fully meshed networkwith 20 links in each trunk to illustrate by how muchlonger the holding time of long-lived traffic should be thanthat of short-lived traffic, for the quasi-stationary approxi-mation to be accurate. The result is shown in Fig. 12. In thescenario we consider, both the offered long-lived traffic andthe offered short-lived traffic are 3.6 Erlangs, and themeanholding time of long-lived calls (1∕μ1) is 1.

As expected, when the holding times of short-lived calls(1∕μ2) are larger than or close to those of long-lived calls,the short-lived calls cannot reach steady state while thenumber of long-lived calls remains unchanged, so thequasi-stationary approximation is inaccurate. However,when the holding times of short-lived calls are significantlyshorter than those of long-lived calls (e.g., by more than 2orders of magnitude, namely, μ2 > 100), the blocking prob-ability for the short-lived traffic becomes invariant to fur-ther increase in the ratio μ2∕μ1, indicating that short-livedtraffic may approximately reach steady state while itsbackground state (due to long-lived traffic activities) re-mains approximately unchanged, so the quasi-stationaryapproximation can lead to accurate results. The errors

shown in Fig. 12 in this condition are mainly due to theoverflow error and path error discussed above. From thefigure, we can observe that our approximation methodswork well when the average holding times of short-livedcalls are less than 5% of the average holding times oflong-lived calls.

I. Effect of the Shape of the Holding TimeDistribution

The results presented above are based on theassumption that the holding times of both the long-livedand short-lived traffic are exponentially distributed. It istherefore important to examine the robustness of the ap-proximations to the shape of the holding time distribution.To this end, we compare the results obtained under the ex-ponential assumptions versus the results obtained underheavy-tailed holding time distribution, where we maintainthe same mean for the two alternatives.

In particular, we consider our heavy-tailed holdingtimes, to follow a Pareto distribution with a comple-mentary distribution function presented in SubsectionH.1 in [28], with δ (seconds) being the scale parameterand minimum holding time and γ being the shapeparameter.

In our simulation we set δ � 0.5 for long-livedtraffic, δ � 0.0025 for short-lived traffic, and γ � 2 forboth types of traffic. All the other parameters are keptthe same as in the default parameter setting inSubsection IV.A.

Figure 13(a) shows the simulation blocking probabilityfor the long-lived traffic (single priority) for a 6-node fullymeshed network with holding time exponential distributedand Pareto distributed. The two curves are very close toeach other, and their confidence intervals are overlapped,which shows that the blocking probability for the long-livedtraffic (single priority) is insensitive to the holding timedistribution.

Figure 13(b) shows the simulation blocking probabilityfor the short-lived traffic for a 6-node fully meshed networkof three cases:

Fig. 11. Blocking probability for short-lived traffic in a 6-node fully meshed network with long-lived and short-lived traffic. The ratio ofthe offered long-lived traffic to the offered short-lived traffic is (a) 1∶1 and (b) 5∶1.

Fig. 12. Average blocking probabilities for the short-lived trafficwith different values of μ2 for a 6-node fully meshed network with20 links in each trunk and μ1 � 1.


1. exponential exponential—holding time of both long-lived and short-lived calls are exponentiallydistributed,

2. exponential Pareto—holding time of long-lived traffic isexponentially distributed while that of short-lived traf-fic is Pareto distributed,

3. Pareto Pareto—holding time of both long-lived andshort-lived calls are Pareto distributed.

The closeness of the simulation results of the three casesillustrates that the blocking probability for the short-livedtraffic for a 6-node fully meshed network is insensitive tothe holding time distribution shape.

We have produced equivalent simulation results for thecase of the NSF network topology. The results are pre-sented in Figs. 14(a) and 14(b). We observe similar behavior

as in the case of fully meshed networks that provide furtherevidence that the assumption of exponential holding timedistribution is reasonable and that the blocking probabilityis not very sensitive to the shape on the holding timedistribution.

Fig. 13. Average blocking probabilities for (a) long-lived traffic (single priority) and (b) short-lived traffic, considering different servicetime distributions for a 6-node fully meshed network with 20 links in each trunk.

Fig. 14. Average blocking probabilities for (a) long-lived traffic (single priority) and (b) short-lived traffic, considering different servicetime distributions for the NSF network with 20 links in each trunk.

TABLE IIICOMPUTATIONAL COMPLEXITY OF THE ALGORITHMS

Trunk Capacity Running Time of EFPA (s) Running Time of OPCA (s) Memory of EFPA (bytes) Memory of OPCA (bytes)

C � 20 0.173369 0.413720 66840 89040C � 100 0.431417 3.034318 95640 137040C � 1000 278.515913 1065.012123 419640 677040

Fig. 15. Coronet topology.


These results are consistent with equivalent resultsobtained in [28] for optical burst switching networks.

J. Computational Complexity of the Algorithms

It is difficult to provide general analytical results for thecomputational complexity of OPCA and EFPA becauseboth require fixed-point iterations to converge. Never-theless, we provide numerical examples that illustratethe time and memory complexity of the algorithms.

In Table III, we provide information on running timesand memory usage of both EFPA and OPCA required forthe network blocking probability computation in a six-nodefully meshed network for three cases representing differenttrunk capacity values.

Wehaveobserved fromthesenumerical examples that forasmallnetwork,OPCArequiresmorecomputationtimeandmorememory thanEFPA, but the overall computing resour-ces are manageable. As we demonstrate in the next subsec-tion, OPCA is also applicable to the large Coronet networkwhile the equivalent EFPA results are unattainable.

K. The Coronet

We demonstrate here that OPCA is applicable to largescale networks such as the Coronet, shown in Fig. 15, whilesimulation results are computationally prohibitive for suchlarge scalenetworks.Withall of the9900SDpairs in thenet-work and one alternative path for each SDpair, the networkblocking probabilities for long-lived and short-lived trafficcan be obtained by OPCA within reasonable running time,as shown inFig. 16. The number of links per trunk is 50, andthe thresholds for long-lived traffic and short-lived trafficare 40 (80%) and45 (90%). Themaximumallowable numberof alternate paths is 1. All the other parameters are kept thesameas in the default parameter setting inSubsection IV.A.The running times used to calculate the network blockingprobabilities in theCoronet is about 121.734439 s byOPCA,obtained using MATLAB 7.6.0 executed on a desktop PC

with IntelR CoreTM 2 Quad at 3 GHz CPU, 4 GHz RAM,and 32 bit operating system.

V. CONCLUSIONS

In this paper, we have considered a circuit-switchednetwork with long-lived and short-lived calls where thelong-lived calls can preempt the short-lived ones. We useEFPA and OPCA combined with the quasi-stationaryapproximation to estimate the blocking probabilities.The results demonstrate that in most cases, OPCA can es-timate the blocking probabilities reasonably well, andgenerally, better than EFPA. As long-lived calls providebackground traffic for short-lived ones, the ratio of theiroffered load also affects the accuracy of the approximations.Reduction of offered long-lived traffic together with anincrease of offered short-lived traffic will improve the accu-racy of OPCA, while that of EFPA is not much improved.However, when the number of links on each trunk in-creases, the performance of EFPA is improved. Allowingmore alternate path traffic, either by increasing the maxi-mum allowable alternate paths or the long-lived trafficthreshold, is beneficial under light traffic. However, whenthe network is fully occupied, it is important to restrict al-ternate path traffic. We have observed that the quasi-stationary approximation requires that the mean holdingtime of long-lived calls is at least 20 times longer than thatof short-lived calls. Nevertheless, this is not a very restric-tive requirement if long-lived calls represent static callsand short-lived calls represent dynamic ones. We have alsodemonstrated that approximating blocking probabilitybased on the exponential holding time assumption is notvery sensitive to the shape of the holding time distributionand is fairly accurate also for heavy-tailed holding time dis-tributions. We have illustrated by numerical examplesthat, for a small network, OPCA requires more computa-tion time and more memory than EFPA, but the overallcomputing resources are manageable. For a large scale net-work such as the Coronet, we have demonstrated thatOPCA is also applicable while the equivalent EFPA resultsare unattainable.

Fig. 16. Blocking probabilities obtained by OPCA for (a) long-lived traffic (single priority), and (b) short-lived traffic for the Coronet with50 links per trunk.


ACKNOWLEDGMENTS

Thework described in this paperwas partially supported bya grant from the Research Grants Council of theHongKongSpecial Administrative Region, China (CityU 123012) andbyagrant fromCityUniversity ofHongKong (No. 7002860).

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Meiqian Wang received her Bachelor’s de-gree and Master’s degree from the Depart-ment of Electronic Engineering at HarbinInstitute of Technology in 2006 and 2009, re-spectively. She is now working toward herPh.D. degree at the City University of HongKong. Her research interest lies in perfor-mance evaluation in circuit switching andburst switching networks.

Shuo Li received the B.Sc. degree in elec-tronic and communication engineering fromthe City University of Hong Kong, HongKong, in 2009. She is currently working to-ward her Ph.D. degree in the Department ofElectronic Engineering at the City Univer-sity of Hong Kong. Her research interestsare performance evaluation of telecommuni-cation networks.

Eric W. M. Wong (S’87–M’90–SM’00) re-ceived the B.Sc. and M.Phil. degrees in elec-tronic engineering from the ChineseUniversity of Hong Kong, Hong Kong, in1988 and 1990, respectively, and the Ph.D.degree in electrical and computer engineer-ing from the University of Massachusetts,Amherst, in 1994. In 1994, he joined the CityUniversity of Hong Kong, where he is nowan Associate Professor with the Department

of Electronic Engineering. His research interests include theanalysis and design of telecommunications networks, opticalswitching, and video-on-demand systems.

Moshe Zukerman (M’87–SM’91–F’07) re-ceived the B.Sc. degree in industrial engi-neering and management and the M.Sc.degree in operations research from theTechnion–Israel Institute of Technology,Haifa, Israel, and the Ph.D. degree in engi-neering from the University of California,Los Angeles, in 1985. He was an indepen-dent consultant with the IRI Corporationand a Postdoctoral Fellow with the Univer-

sity of California, Los Angeles, in 1985–1986. In 1986–1997, hewas with the Telstra Research Laboratories (TRL), first as a Re-search Engineer and, in 1988–1997, as a Project Leader.He also taught and supervised graduate students at MonashUniversity in 1990–2001. During 1997–2008, he was with TheUni-versity of Melbourne, Victoria, Australia. In 2008 he joined theCity University of Hong Kong as a Chair Professor of InformationEngineering and a team leader. He has served on various journaleditorial boards and conference technical program committees.


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