A CRITICAL REVIEW OF A STEVENS’ AND SINCLAIR’S EQUATION

Emeka D. OkaekwuSchool of Electrical and Information Engineering

University of the Witwatersrand, Johannesburg

emeka.okaekwu@yahoo.com

Nicholas AdeSchool of Physics

University of the Witwatersrand, Johannesburg

leronicholson@yahoo.ca

24 September 2009

Outline

Abstract

Introduction

Aim / Objective

Methodology

Results

Discussion

Conclusion

Abstract A finite-source queuing model is very useful in determining

performance equations, the 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.

 

Introduction Modern telecommunication systems experience

heavy sources of traffic especially at the busy hour. 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. Systems operating on delay basis require an equation for the probability of delay in an M/M/r/K/K 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. Boucher [2] pointed out that such a delay probability was missing

Introduction (cont.)

Three basic teletraffic formulae have been in use in telecommunications systems: the Erlang-B , Erlang-C and Engset formulae. The Erlang’s formulae are used in systems (blocking systems) with infinite sources of traffic while the Engset formula is used for systems (delaying systems) with a finite population. No delay probability equation for a system with a finite population has been used. This brings us to the concept of Boucher’s decision tree [2] where the delay formula for finite number of sources is missing.

Table 1. THE THREE BASIC TRAFFIC FORMULAE AND BOUCHER’S MISSING

FORMULA

Blocking systems

Calls lost

Queuing Systems

Calls delayed

Infinite population Erlang-B Erlang-C

Finite Population Engset

??

Aim / Objective

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.

Methodology

A mathematical approach is used to analyze Stevens’ and Sinclair’s performance equation. The performance equation is denoted Eq. (SS) and we shall refer to Eq. (12) in Stevens’ and Sinclair’s paper [1]. The mathematical 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. The entire analysis was then tested with a MATLAB computer program.

Erlang-C Formula Erlang-C Formula:

Where, Pc denotes the blocking probability for N busy servers, A is the traffic load, i is a counter for the summation.

  

[(AN / N!) * (N/N-A)] Pc = ----------------------------------------- (1)

1

0

!/N

i

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

Equation (SS) Eq. (SS):

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)

Definition of Parameters in Eq. (SS)

β = α/µ, the busy to idle ratio of each source, which is the offer load in Erlangs.

Pd = probability of delay 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 (service rate) α = call arrival rate (call origination) n = counter for the summation

Analysis Eq. (SS) and Erlang-C were computed using

the the following Parameters, β = 2, K = 10, r = 5.

In the analysis, care was taken in choosing values of offered traffic (β ), number of servers (r ) and number of sources (K ) to allow for ease of analysis and to avoid numeric overflow in computation.

Results The probability of delay (Pd) computed from Eq.

(SS) was 0.04 (4%) and that obtained from Erlang-C equation was 0.05 (5%) which is actually greater than 4% thus implying a certain degree of satisfactory performance.

The MATLAB program was executed with the same 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.

Discussion A delay probability formula for finite sources of

traffic is available. Table 1 is now complete as shown below.

The Four Basic Teletraffic FormulaeBlocking systems

Calls lost

Queuing Systems

Calls delayed

Infinite population

Erlang - B Erlang - C

Finite Population

Engset Equation SS

Conclusion Stevens’ and Sinclair’s equation is

actually the DELAY FORMULA for finite sources of traffic missing in Boucher’s decision tree[2] as the results of the analysis prove that the equation is useful in determining the queuing probability for finite sources.

Thank you

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.