ON A MULTI-CHANNEL QUEUE WITH STATE DEPENDENT INPUT FLOW AND INTERRUPTIONS

This paper deals with a multi-channel queueing system with a finite waiting room but without losses. The latter is achieved by a temporary interruption of the input flow activity until the waiting room is ready to place a new customer. In addition, the input flow on its "busy period" is non-recurrent: It is state dependent and may be controlled over relevant times of decision making. A similar model without interruptions (i.e. with losses) was earlier studied by the author, where in particular, major probability characteristics of the queueing process in equilibrium were obtained. Now the author derives a simple explicit relation between the two models allowing the given queue to inherit the results previously obtained. New techniques for semi-regenerative processes are used.


Introduction
We consider a modification of the queueing model treated in Dshalalow (1985).In this m-channel model with the finite waiting room, the input stream was controlled at the times when customers depart from the "source" subject to the number of customers in the system.One of the goals in studying the model was to reduce losses of customers entering a 1Received: May 1989, Revised: September 1989.fully loaded system.In a transportation servicing system this would mean that a transport unit (i.e.customer) leaves the source with the slowest possible rate whenever the waiting room is fully occupied.However, even this system can not completely eliminate losses that are usually detrimental to any real system.In the present model we do exclude losses by bringing the input stream to the full stop when the number of customers in the servicing system attains a certain critical level.This system is somewhat more difficult in treatment compared with the original model, in the same way as queues with non-recurrent and recurrent inputs.But fortunately there is a trick how to connect the both models with a simple relation.
We consider queueing processes in the following two versions of m-channel systems with interarrival times of the input process dependent on the number of customers.There is a limited number of waiting places in both systems.The first system maintains the service of customers that do not even depart from the source if the waiting room is full, until a place in the waiting room is free again.We call this system a queue with flow interruptions.
There is no such model (at least as a multi-channel one) in the existing literature known to the author.The second system admits losses whenever it is full.It has been mentioned that the control over the input process can indeed reduce such losses but not completely eliminate them.One of the negative sides of this "low-risk-policy" is that the system usually takes measures against losses by reducing the input rate in advance sucrifising the total effectiveness of system output.
A simple relation between the probability distributions of the queueing process in the steady state is derived by applying techniques for semi-regenerative processes.This is given in terms of an explicit formula.The result obtained will surely be of interest also for multi-channel queues of type GI/M/m/N with no control over the input stream.

Formal Description
The input stream of single customers is described by the point process n=l,2,...}.The arriving customers are served by any of m free parallel channels available according to the FIFO discipline.Otherweise, the customers take places in the waiting room (of a finite capacity).Let {Q(t); t_>0} denote the number of customers in the system.We set for model 1 (with interruptions) the capacity of the waiting room w+ 1 and for model 2 (no interruptions) w assuming that w>_0.In case of the total loss" queue for both systems, t_>0) is semi-regenerative relative to the sequence { Tn}.Consequently, the imbedded process (f,l,(P),l ,(Qn)) --* E is a Markov chain (MC).Since the periods of interruptions are distributed exponentially, it is easy to see that the MC's in both models are stochastically equivalent and they are obviously ergodie.Let P:.=(Po,P1,...,Pm+w) be the invariant probability measure of the MC (Qn) for both models.Therefore, the limiting probability r: (r, r = ,...,rm+w+l) (ease of model 1 with interruptions), rZ:=(r0,r,...,rm+,) (ease of model 2 without interruptions) of the semi-regenerative process (Q(t)) exists (in general under the same conditions as P does).For the next we agree to lable with superscripts 1 or 2 corresponding probability characteristics of both models whenever their difference should be empfasized.
(3.1) with Let (f, I, (P)zd ,(Y; t>_0)) --, (0,1,...,m+ w) be the semi-Markov process associated with the sequence of stopping times { Tn}.This is also ergodic and its limiting probabilities can be expressed through P (cf.inlar (1975), p. 342).We need this expression for the queue with interruptions.Let Y be the limit of Y in the sense of the weak convergence and let =(Y0, Yl,--.,Ym+w)be its probability ditsribution.We now add to y's the upper index 1 in accordance with the above convention.Then Y = PM k=0,1,...,m+w, (3.2) M,:= E[T1] = ( a k=O,1,...,m+w-1 am+w+ k=m+ w and PM is the scalar product of P and M= {M0, M1,... ,M,n+w } r equals (3.3) PM= PA + P,n+w 1 In the last expression, PA denotes the scalar product of P and A:=(ao, a,...,a,n+w }7. Let , and , be the nth busy period of the input stream and the interruption period following the nth busy period, respectively, in case of model 1.Let {; t>_0} be the counting process associated with the point process {fin; n=l,2,...}.It is readily seen that the processes Q2(t) and Q(t) on its busy periods are stochastically equivalent or formally it holds