A CRITICAL REVIEW OF STEVENS’ AND SINCLAIR’S EQUATION

Emeka D. Okaekwu School of Electrical and Information

Engineering University of the Witwatersrand, Johannesburg

emeka.okaekwu@yahoo.com

Abstract: A finite-source queuing model is very useful in determining performance equations, probability of delay of users who queue for a free server when all servers are busy. The aim of this paper is to critically examine and analyze Stevens’ and Sinclair’s equation to show that it is a performance equation for the probability of delay in private mobile radio (PMR) systems. We analyze such an equation mathematically and present a computational program using MATLAB. The system is an M/M/r/K/K queuing model with finite sources of traffic. Some of the basic traffic characteristics and assumptions are also reviewed. Key words: blocking probability, delay probability, Erlang-C formula, infinite population, finite population, grade of service (GoS), full availability systems.

I. INTRODUCTION Modern telecommunication systems experience heavy sources of traffic especially at the busy hour. This requires that the GoS for offered traffic be improved efficiently to satisfy network subscribers. In queuing systems, subscribers are queued until they are served. If they are made to wait for too long, they may lose patience and default from the queue, resulting in no service being offered. The general traffic problem is therefore to calculate the probability that a call attempt is blocked because of unavailable traffic resource (servers). On the other hand, systems operating on delay basis require an equation for the probability of delay in an M/M/r/KK model. Such an equation is available for a system with an infinite population, but no such equation has been used for a system with finite sources of traffic.

Nicholas Ade

School of Physics University of the Witwatersrand, Johannesburg

leronicholson@yahoo.ca II. HISTORICAL REVIEW Three basic teletraffic formulae have been in use in telecommunication systems: the Erlang-B and Erlang-C formulae, and the Engset formula. The Erlang-B and Engset formulae are used in blocking systems to calculate the probability that a call attempt will be blocked or lost, whereas the Erlang-C is used in queuing systems to find the probability that a call attempt will be delayed. Furthermore, the Erlang’s formulae are used in systems with infinite sources of traffic while the Engset formula is used for systems with a finite population. No delay probability equation for a system with a finite population has been used. Boucher [2] pointed out that such a delay probability was missing. Miller [3] established a delay equation (Eq. {30b} in [3]) for a finite population, but his assumption was that of blocked calls held instead of the assumption of blocked calls delayed for the disposition of blocked calls and so his Eq. (30b) is not the delay probability equation missing in Boucher’s decision tree [2] as discussed further below. From the point of view of [1], the nature of traffic is mostly from a finite number of sources since communication is usually one-to-many, with group members being party to calls. This is usually found in private mobile radio systems. Over the years, many models and formulae have been proposed that could cater for infinite and finite number of sources for different call dispositions like for cleared, held and delayed calls. This brings us to the concept of Boucher’s decision tree [2] where the delay formula for finite number of sources is missing. This forms our basis of analysis of a performance equation for the queuing probability for finite number of sources. The performance equation is denoted Eq. (SS) and we shall refer to Eq. (12) in Stevens’ and Sinclair’s paper [1].

Table 1 shows the various formulae for finite and infinite population. TABLE 1[6] THE FOUR BASIC TRAFFIC FORMULAE

Blocking system

Calls lost

Queuing system

Calls delayed

Infinite population

Erlang-B Erlang-C

Finite population

Engset Eqn. (S-S)

III. DISPOSITION OF BLOCKED CALLS Boucher [2] exposed some hidden details concerning the disposition of blocked calls. The three common cases are:

If an idle server is not immediately available, the source becomes idle since the call is cleared from the system. This is the blocked calls cleared assumption.

If an idle server is not immediately available, the call is held for an interval equal to its holding time, and then the source becomes idle. If an idle server becomes available during the waiting period, it will be seized and held for an interval equal to the remaining portion of its mean holding time. This is the blocked calls held assumption.

If an idle server is not available, the call is

queued until an idle server is available. When an idle server becomes available, it will be seized to serve the next call in the queue and held for the full call holding time. This is the blocked calls delayed assumption.

The third assumption gives a clear insight of what a delay formula is and carefully makes equation (30b) of Miller’s equation [3] irrelevant as far as the context of delay formula is concerned.

1V. GRADE OF SERVICE Grade of service (GoS) is defined as the probability that offered traffic will be blocked or delayed [3]. GoS for blocking systems is expressed as the

probability of loss. For delay systems, it is expressed in terms of the probability of delay exceeding t seconds and average delay of offered or delayed calls [4]. In every traffic problem, there are three interrelated parameters: offered traffic, traffic-handling resources (servers), service objective (GoS). The offered load (traffic), number of servers and the underlying traffic assumptions and characteristics in turn determine the GoS [4]. A server’s GoS is actually a prediction of the probability of congestion (i.e. the probability that a call is blocked or delayed) at a given level of offered load, and not an absolute one. For instance, a GoS of 0.02 does not mean that exactly two calls in a hundred will be blocked or delayed during the busy hour. It rather means that given a large volume of traffic, the probability of congestion will tend toward two in a hundred. A GoS of one implies an absolutely blocking system. This means the closer the GoS is to zero, the better the system. V. FULL AVAILABILITY SYSTEMS In a switching full availability, any inlet can be connected to any outlet; otherwise we have a limited availability system. When a connection cannot be made in a switching system because of all outlets being busy, the condition is known as congestion. In a state of congestion, an incoming call can be blocked (a loss system) or it can wait for a connection (a waiting or delay system). Delay systems are queuing systems operating on a delay basis such that when no traffic resource is available, a new call arrival waits in a queue until a resource becomes available. Full availability systems are described by the following [5]:

Input process: describes ways of call origination or arrival. We consider Poisson arrival with random call origination. The Poisson distribution [3,5] has mean λt, where λ is called the arrival rate or termination rate.

Service mechanism: describes the number of outlets (e.g. trunks, servers, holding time distribution, etc). The exponential distribution applies to telephone traffic.

Queue discipline: specifies ways of handling calls during congestion (loss or delay). In delay systems (our target system), we must specify the order of serving waiting calls. The order may be first-in first-out (FIFO), last-in first-out (LIFO), random service order (RSO), etc.

Classification of full availability systems uses the Kendall notation: A/B/s where A represents

distribution of input process, B service time distribution and s the number of servers. For A and B, the following symbol is used: M, exponential (Markov). M is just one of a set of symbols. M suffices for use in the present discussion. As an example, a system with Poisson arrival, exponential service time and s servers are expressed as M/M/s. With n finite sources, we have M(n)/M/s, and with waiting room of m positions, we have M/M/s(m) or M/M/s+m. A loss system is therefore expressed as M/M/s(O) or M/M/s/s. Without such additional notations, a FIFO system with infinite waiting room is meant [5]. VI. THE ERLANG-C EQUATION The Erlang-C equation is used to calculate the blocking probability for an M/M/N/K/K queue where subscribers who find all N traffic resources busy are not lost, but are delayed in the queue with an infinite number of K/K queuing space (waiting room). The Erlang-C equation is given by: [(AN / N!) * (N/N-A)] Pc = ----------------------------------------- (1)

1

0!/

N

i

i iA + [(AN / N!) * (N/N-A)]

Where, Pc denotes the blocking probability for N busy servers, A is the traffic load, i is a counter for the summation. We observe that Eq. (1) is recursive because of the factorial (!) function. VII. MATHEMATICL ANALYSIS OF STEVENS’ AND SINCLAIR’S EQUATION Eq. (SS) A simplified mathematical analysis of Eq. (SS) is presented. The analysis is then compared with Erlang-C system and deductions made with respect to the definiteness of the Eq. (SS) in catering for the queuing probability for finite number of sources. In analyzing Eq. (SS) mathematically, care is taken in choosing values of offered traffic, number of servers and number of sources to allow for ease analysis and to avoid numeric overflow in computation. Eq. (SS), which we refer to, is Eq. (12) in [1]:

Pd = β/Ns )( nKPK

rnn

------------------ (2)

Where,

Ns =

1

1

r

nnnP +

K

rnnrP -------------------- (2.1)

Pn = Dn/D -------- (2.2)

Dn = [K! / (K-n)!] βn/n!, 1 < n ≤ r ------- (2.3) Dn = [K!/(K-n)!] βn/r!rn-r , r < n ≤ K ---- (2.4) β = α/µ, the busy to idle ratio of each source, which is the offer load in erlangs. Pd = probability of delay > µ Ns = mean number being served (the carried traffic) r = Number of servers in full-availability K = Number of sources h = Average holding time µ = acceptable delay α = call arrival rate (call origination) n = counter for the summation. Given the following Parameters, β = 2, K = 10, r = 5, we compute Eq. (2.1) as follows: Pn = Dn/D Where, Dn = [K! / (K-n)!] βn/n! for 1 < n ≤ r Dn = [K!/(K-n)!] βn/r!rn-r for r < n ≤ K D = D0 + D1 + D2 + …………..+ D10

D0 = 1 D1 = kβ = 10*2 = 20 D2 = [10! / 8!] * 22 / 2! = 180 D3 = [10! / 7!] * 23 / 3! = 960 D4 = [10! / 6!] * 24 / 4! = 3360 D5 = [10! / 5!] * 25 / 5! = 5064 D6 = [10! / 4!] * 26 / 5! * 5 = 403,200 D7 = [10! / 3!] * 27 / 5! * 52 = 16,128,000 D8 = [10! / 2!] * 28 / 5! * 53 = 483,840,005 D9 = [10! / 1!] * 29 / 5! * 54 = 9,676,800,000 = 9.6768 exp 9 D10 = [10! / 0!] * 210 / 5! * 55 = 9.6768 exp10 But D = D1 + D2 + …+ D10 = 1 + 20 + 180 +…+ 9.6768 exp 10 = 1.0694 exp 11 But Pn = Dn/D, n = 1,…10

P0 = D0 / D = [1 / 1.0694 exp 11] = 9.3505 exp -12 P1 = D1 / D = [20 /1.0694 exp 11] = 1.870 exp -10 Similarly, we have: P2 = D2 / D = 1exp -9 P3 = D3 / D = 8 exp -8 P4 = D4 / D = 3.1 exp- 8 P5 = D5 / D = 7.5 exp -6 P6 = D6 / D = 3.77 exp -4 P7 = D7 / D = 1.5081 exp -3 P8 = D8 / D = 4.5244 exp -3 P9 = D9 / D = 9.048 exp -2 P10 = D10 / D = 0.9048

Ns =

1

1

r

nnnP +

K

rnnrP

4

1nnnP = P1+ 2P2 + 3P3 + 4P4

= 1.5 exp -7

K

rnnrP =

10

55

nnP = 5P5 + 5P 6 + 5P7 + 5P8 +

5P9 + 5P10 = 5(P5 + P6 + P7 + P8 + P9 + P10) = 5(0.9999) = 4.9997

Ns =

1

1

r

nnnP +

K

rnnrP

= (1.5 exp-7) + 4.9997 ≈ 4.9997

Then, )( nKPK

rnn

=

10

5)10(

nn nP

= P5(10-5) + P6(10-6) + P7(10-7) + P8(10-8) + P9(10-9) + P10(10-10)

= 5P5 + 4P6 + 3P7 + 2P8 + P9

= 3.75 (exp -7) + (1.508 exp -5) + (4.524 exp -4) + (9.048 exp -3) + (9.048 exp- 2)

)( nKPK

rnn

= 0.0999

Pd = β/Ns )( nKPK

rnn

= (2 / 4.9997) * 0.0999

= 0.0399 For Erlang-C system (Eq. (1)), we have [(AN / N!) * (N/N-A)] Pc = ------------------------------------

1

0!/

N

i

i iA + [(AN / N!) * (N/N-A)]

Where, A = β = 2 and N = r = 5.

Then,

1

0!/

N

i

i iA =

4

0!/

i

i iA

= (20 / 0!) + (21/ 1!)+ (22 / 2!) + (23 / 3!) + (24 / 4!) = 1 + 2 + 2 + (8 / 6) + (16 / 24) = 7 Also, [(AN / N!) * (N / N-A)] = (25 / 5!) * (5 / 5-2) = (32 /120) * (5 / 3) = 0.4444 Therefore Pc = (0.4444) / (7+0.4444) = 0.0597 With the above input parameters and comparing the figures for both SS and Erlang-C, the value of the probability of delay, Pd for Eq. (SS) is 0.04 = 4%, and that obtained from Erlang-C equation is 0.05 = 5%, which is actually more than 4% thus implying a certain degree of satisfactory performance. We can see that the value of the SS equation is less than that of Erlang-C is an ideal equation that can cater for the queuing probability of finite sources.

VIII. COMPUTER PROGRAM USING MATLAB clear; clc; beta = 2; K = 10; r = 5; D0 = 1; for (n=1:r) D(n) = (factorial(K) / factorial(K - n)) * (beta^n) / factorial(n); end for (n=(r+1):K) D(n) = (factorial(K) / factorial(K - n)) * (beta^n) / factorial(r) * r^(n-r); end sumD = sum(D) + D0; Mn = 0; for (n=1:(r-1)) Mn = Mn + (n * ((D(n) / sumD))); end Ac = 0; for (n=r:K) Ac = Ac + (r * ((D(n) / sumD))); end Ns = Mn + Ac; Pd1 = beta / Ns; Pd2 = 0; for (n=r:K) Pd2 = Pd2 + (((D(n) / sumD)) * (K - n)) ;

end Pd = Pd1 * Pd2 For Erlang-C, the program is: clear; clc; A = 2; N = 5; X = 0; for (i=0:(N-1)) X = X + ((A^i) / factorial(i)); end Y = ((A^N)*N) / (factorial(N) * (N-1)); Pc = (Y / (X + Y)) IX. DISCUSSION The MATLAB program above was tested and run with the input parameters β = 2, r = 5 and k = 10. For Eq. (SS), it gives a value of 0.0400 which is the same as what was obtained in the mathematical analysis and Erlang-C gives a value of 0.0455 which is also equivalent to what was obtained earlier. Stevens’ and Sinclair’s equation is an application of a full availability system using the Kendall notation. The equation considers K equal sources, each being a group of mobile users and a dispatcher forming a net attempting to use one of r radio channels, with FCFS (FIFO) employed. In Eq. (SS), it is assumed that K > r for there to be a delay and the last state is K because if K ≤ r, there will be no delay since all calls would be served immediately [1]. This is exactly an M/M/r/K/K queuing system model, where the K/K denotes sufficient queuing space (waiting room) if all K sources are in the system. The M/M/r/K/K model has upward rates as multiples of the number of idle sources and the inverse of their idle time. Guides for the selection of traffic formulae can be obtained from a table and a diagram. The table is called traffic formula guide and the diagram is called traffic formula decision tree [2]. All the main traffic formulae given in [2] are based on these guides so any other ideal formula should follow the guides. In [2], the decision tree has a vacancy for the delay

formula for finite source of traffic. Using the diagram and the selection guide table, we now incorporate Equation SS in the selection guide table as follows:

Traffic Formula: = Equation SS Holding Time Distribution: = Exponential Blocked-call Disposition: = Delayed Number of Sources: = Finite Typical Applications: = Used for planning

PMR systems which employ FCFS queuing for r radio channels. PMR systems are used by organizations such as the emergency services, public utilities, local authorities, and public transport [1].

X. CONCLUSION The above analysis on Stevens’ and Sinclair equation depicts that the equation is actually the DELAY FORMULA for finite sources of traffic missing in [2] as the mathematical analysis proves that equation SS is useful in determining the queuing probability for finite sources. The values obtained from the mathematical approach are ideal as they fall in line with teletraffic concepts, theory and simple reasoning. The detailed theoretical analysis also shows that the equation follows all the teletraffic concepts in a logical manner. REFERENCES [1] Stevens, R.D., Sinclair, M.C. “Finite- source analysis of traffic on private mobile radio systems”, Electronics letters, 17th July 1997, Vol. 3, No.15, pp 1292 - 1293. [2] Boucher, J.R., “Traffic system design handbook”, IEEE Press, 1993, pg 11. [3] Miller, L.E., Formulas for Blocking Probability, April 2000, pp 1 – 12. [4] Bernhard E.Keriser, and Eugene Stranger: digital telephony and Network Integration, 2ndedition, Van Nostrand Reinhold, New York, 1995, pp 617 – 619 [5] Akimaru, K. and Kawashima, K., Teletraffic Theory and Applications, 2nd edition, Springer, 1997, pp 3 - 10.

[6] Kennedy, I.G., and Krzesinski, A.E., The recursive evaluation of the M/M/r/K/K queue, http://home.intekom.com/satnac/proceed Ings/2008/papers/planning/Kennedy%20 No%2019.pdf, 2008, pg 1. Emeka D. Okaekwu is an M. Eng. Student at the School of electrical and Information Engineering, University of the Witwatersrand, Johannesburg. Nicholas Ade is currently an MSc student at the School of Medical Physics, University of the Witwatersrand, Johannesburg.