A Basically Poisson Queue with Non-Poisson OutputAuthor(s): N. HadidiSource: Advances in Applied Probability, Vol. 6, No. 2 (Jun., 1974), pp. 254-255Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426289 .

Accessed: 13/06/2014 22:35

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Applied Probability Trust is collaborating with JSTOR to digitize, preserve and extend access to Advances inApplied Probability.

http://www.jstor.org

This content downloaded from 185.44.77.82 on Fri, 13 Jun 2014 22:35:51 PMAll use subject to JSTOR Terms and Conditions

254 3RD CONFERENCE ON STOCHASTIC PROCESSES AND APPLICATIONS

A basically Poisson queue with non-Poisson output

N. HADIDI, Pahlavi University, Shiraz, Iran

It has been shown by Burke [1] (among others) that the output interval defined as the time interval separating two consecutive departures--after service com- pletion-from the system in the queueing model M/M/1 in the steady state follows the inter-arrival distribution. Mirasol [7] has shown that this result is also true for the more general model M/G/oo, though his argument needs to be augmented in accordance with Shanbhag [8]. This has further been shown, by the author [2], to be true for the basically Poisson model M/Mn/1 in which the otherwise Poisson stream of service completions has parameter ny when the system contains n customers. Other results on such a model are found in Hadidi and Conolly [6], Hadidi [3] and [4].

This model has originally been considered in view of its evident reduction in congestion as explored in the above references. An obvious counter-part to it is the model Mn/M/1 where here the otherwise Poisson stream of arrival epochs has parameter A/(n + 1) when the system contains n customers. Results on this model are found in Hadidi [4] and [5]. While the arrival pattern is basically Poisson, we shall show that the steady state output intervals are not negative exponentially distributed and consecutive output intervals are stochastically dependent.

The joint density of two consecutive output intervals is given by

h(t, u) = p2e-e(t+u)

1+ 1((p - 1) +-2 + 1)e-At( - (pe-(U)u

- pe-(A-)'(1 -2e-(- ,}

where p is the service parameter and p = )/y. This implies the probability density

g(t) = pe-"t + (p-e'"- Ae- (p - 1)(eP - 1)

for a single output interval and a cof.Tizient of correlation

eP(2 - p2) - 2(1 + p) [pe2P + 2eP(1 -p) -2](p +2)

between two consecutive intervals.

References

[1] BURKE, P. J. (1956) The output of a queueing system. Operat. Res. 4, 699-707. [2] HADIDI, N. (1972) On the output process of a state dependent queue. Skand. Aktuartidskr.

10, 182-186. [3] HADIDI, N. (1974) Busy period of Poisson queues with state-dependent arrival and

service rates J. Appl. Prob. 11, Nol 4. To appear.

This content downloaded from 185.44.77.82 on Fri, 13 Jun 2014 22:35:51 PMAll use subject to JSTOR Terms and Conditions

Sheffield, 13-17 August 1973 255

[4] HADIDI, N. (1969) On the service time distribution and waiting time process of a potent- ially infinite capacity queueing system. J. Appl. Prob. 6, 594-603.

[5] HADIDI, N. (1973) A queueing madel with valiable arrival rates. Periodica Math. Hungarica. To appear.

[6] HADIDI, N. AND CONOLLY, B. W. (1969) On the improvement of the operational charac- teristics of single server queues by use of a queue length dependent service mechanism. Appl. Statist. 18, 229-240.

[7] MIRASOL, N. M. (1963) The output of an M/G/ co queueing system is Poisson. Operat. Res. 11, 282-284.

[8] SHANBHAG, D. N. (1972) Letter to the Editor. J. Appl. Prob. 9, 470.

Stochastic integral equations applied to telecommunications traffic without delay

P. LE GALL, SOCOTEL, Paris

Modern telecommunications techniques consider the problem of traffic handling in fairly general networks, for traffic without possible delay but with fairly arbit- rary service time distributions. The method of stochastic integral equations al- lows us to tackle and even to solve this problem in an effective and elegant way.

1. One trunk group

In the case of the most general assumptions, Fortet [1] has defined the random function Y(t) of the number of calls in progress at time t, as the solution of the stochastic integral equation:

(1) Y(t) = V[Y(u)] R(u, t)dN(u);

N(t) is the stochastic number of arrivals during the interval [0, t [, the arrival at t excluded. R(u, t) is a stochastic function satisfying the assumptions:

R(u,t)= 1 if u<t<u + T,,

R(ut)= if t<u or t>u+T,,

T, being the random duration of the call starting at time u. V(y) is an algebraic function such that

V(y) = 1 if y = 1,2,...,L- 1,

(0 otherwise

L being the number of trunks in the group. This non-linear integral equation could be solved only when L = 1: see [1].

In [2] the general solution is given, by replacing (1) by the multiple integral equa- tion which defines a random function W,(t):

This content downloaded from 185.44.77.82 on Fri, 13 Jun 2014 22:35:51 PMAll use subject to JSTOR Terms and Conditions