A NON-MARKOVIAN QUEUEING SYSTEM WITH A VARIABLE NUMBER OF CHANNELS

In this paper we study a queueing model of type GI/M/m̃a/∞ with m parallel channels, some of which may suspend their service at specified random moments of time. Whether or not this phenomenon occurs depends on the queue length. The queueing process, which we target, turns out to be semi-regenerative, and we fully explore this utilizing semi-regenerative techniques. This is contrary to the more traditional supplementary variable approach and the less popular approach of combination semi-regenerative and supplementary variable technique. We pass to the limiting distribution of the continuous time parameter process through the embedded Markov chain for which we find the invariant probability measure. All formulas are analytically tractable.


Introduction
This paper analyzes a multi-channel queueing system with a random number of channels, infinite capacity waiting room, general input, and exponentially distributed service times.The total number of channels does not exceed m, but at any given time not all of them are "active."The latter implies that even if a particular channel is busy servicing a customer, the service at some later time can become suspended.We assume that there is a certain sequence {T n } of stopping times relative to the queueing process at which a decision is being made for every busy channel to continue, suspend, or activate service.This policy literally makes the total quantity of servers random and it affects both the servicing and queueing processes.
More formally, if Z t denotes the number of customers in the system at any given time t, the number of active channels at T n (where {T n } is the arrival process) is a binomial random variable with parameters (Z Tn , a) where 0 ≤ a ≤ 1, provided that Z t ≤ m.Unless service is interrupted, each of the customers is being processed at an exponentially distributed time and service durations on each of the active channels are independent.The input is a regular renewal process.For this system we use the symbolic notation GI/M/ ma /∞.
The system, as it is, generalizes the classical GI/M/m/∞ queue by making it, in some real-world applications, more versatile.We can easily associate it with any mail order servicing system where once the order has been taken, it can be suspended at any moment of time for an unaware customer (who believes he is being processed) for various reasons, most commonly due to unavailable items.An item can also be backordered.The company is trying to shop around and find the item, and this takes time and effort.A similar situation occurs in Internet service where jobs are being routinely suspended for numerous reasons by Internet providers.
In this problem setting we make the consequences of this interruption policy milder, in which suspensions take place only if the buffer (or waiting room) is empty, but all or a few channels are occupied.It makes perfect sense to reduce waiting time for many customers from the buffer.
The queueing systems with variable number of channels have been investigated in the past literature.Saati [4] describes such one as a fully exponential system.Most commonly, the queues with unreliable servers (which can break down at any time) and priorities can also fall into this category.Our system is different firstly because it is non-Markovian and secondly because service interruptions do not take place fully arbitrarily, but with some probabilities upon certain random times.The closest problem to ours is the system in Rosson/Dshalalow [3] with no buffer.
In the present paper, we focus our attention on the queueing process, which turns out to be semi-regenerative relative to the sequence {T n } of arrivals.We start with the embedded process over {T n } and turn to the analysis of the queue as a semi-regenerative process, which to the best of our knowledge has not been analyzed this way even in the case of the basic GI/M/m/∞ system, and thus this method is by itself novel for the class of multi-channel queues.We have used this approach in Rosson/Dshalalow [3] for the case of a more rudimentary GI/M/m/0 system with a random number of channels.The paper is organized as follows.Section 2 formalizes the model more rigourously.Section 3 deals with an embedded Markov chain and the invariant probability measure under a given ergodicity condition.Section 4 analyzes the continuous time parameter queueing process followed by an example presented in Section 5.All formulas are obtained in analytically tractable forms.

Description of the system
Let (Ω, F, (P x ) x=0,1,... , Z t ; t ≥ 0) → E = {0, 1, ...} be the stochastic process which describes the evolution of the queueing process in the GI/M/ ma /∞ system introduced in the previous section.In other words, at any time t, Z t gives the total number of units (or customers) in the system including those being in service.The servicing facility has m permanent channels, of which not all are necessarily active.The buffer (or waiting room) is of an infinite capacity, but the system needs to be "watched" for preserving the equilibrium condition.Customers arrive singly in the system in accordance with a standard renewal process {T n ; n = 0, 1, ..., T 0 = 0}.Inter-renewal times have a common PDF (probability distribution function) A(x), with finite mean ā and the LST (Laplace-Stieltjes transform) α(θ).The formation of active or inactive channels is being rendered upon {T n } as follows.If at time T n − (i.e.immediately preceding the nth arrival) the total number of customers is less than m, each of the channels, including the one that is aimed to accommodate the nth customer (just arrived), can become temporarily inactive with probability b.In such a case, the service of a customer by any such channel becomes suspended until the next stopping time T n+1 (of the process Z t ).Every busy channel is active with probability a (= 1 − b) and he is processing a customer a period of time exponentially distributed with parameter µ until T n+1 or the end of service, whichever comes first.Thereby, service times at each of the parallel channels within the random interval [T n , T n+1 ) are conditionally independent, given X n := Z Tn− and the number of active channels is governed by a binomial random variable ξ n with parameters (X n + 1, a) provided that X n < m.If the number of customers in the system upon time T n − is m or more, then all channels are almost surely active until T n+1 .

Notation
We will be using the following notation throughout the remainder of this paper: a = probability of each busy channel at T n+ to be active, n = 0, 1, ... .b = 1 − a = probability of a channel at T n+ to be inactive.

Embedded Process
As a precursor to the key semi-regenerative approach for treating the one-dimensional distribution of the queueing process (Ω, F, (P

Transition Probabilities
Then, where q j+1 s is the probability that s out of j + 1 busy channels are inactive at T n , (1 − e −µx ) j+1−k is the probability that in [0, x], j+1−s−(k−s) customers are processed, and e −µ(k−s)x is the probability that in [0, x], k − s customers are not finished while being treated.Clearly, p jk = 0 for Notice that where and that

The above probabilities (3.1)-(3.4) form the upper block
The lower block of L with the rows from m and all the way down are identical to that for the system GI/M/m/∞: with and The PGF (probability generation function) of the i th row of L 1 is We can easily deduce that L represents an irreducible and aperiodic MC (Markov chain).

The Invariant Probability Measure
According to Abolnikov and Dukhovny [1], for mµα > 1, there exists an invariant probability measure P := (P 0 , P 1 , . ..) of {X n } as a unique positive solution of the matrix equation (3.12) (3.12) also reads For k ≥ m, (3.13), in light of (3.6-3.8),leads to Let us now consider equations (3.16) for k ≥ m.We will seek the solution of (3.16) in the form where A will be evaluated from (P, 1 ) = 1.Inserting (3.17) into (3.16)gives, where δ, according to Takács [6], is a unique, real positive root of (3.19), strictly less than 1 when meeting the ergodicity condition mµα > 1.For k = m, (3.16) and (3.17) yield which, after some algebra and due to (3.19), reduces to To determine the unknowns P 0 , P 1 , . . ., P m−1 let us define the PGF The following proposition gives an equation in U (z) and P i , i = 0, . . ., m − 1, which lead to the derivation of the invariant probabilities.Proof: From (3.13) ,
To obtain P k 's we will use a method similar to that of Takács [6].Given a polynomial function f (z), define the sequence of linear functionals: Since U (z − 1) is identical to the Taylor series of U (z) expanded at 1, applying R r to the polynomial U (z), r = 0, 1, . . ., m − 1, and then multiplying it scalarly by (1, z − 1, . . ., (z − 1) m−1 ) we will thereby reexpand U (z) in a Taylor series at 1 and arrive at its binomial moments: On the other hand, given the binomial moments, we have The following proposition is an analog to the result known for the GI/M/m/∞ queue, except for a r in (3.36) being different.

37)
Proof: Applying R r to (3.25) we have and due to we have where With the usual combinatorics, m−1 i=r−1 or in the form Simplifying, we obtain Denoting and dividing by a r both sides of (3.43), we have and thus .Therefore, In (3.49), for n = 0 and a 0 = 1 On the other hand, by (3.22) and (3.17), Hence, Futhermore, In particular, U 0 = U (1).By (3.52), therefore, Substituting equation (3.53) into (3.55)yields and after substituting equation (3.45) into (3.57)we finally have We are done with the proposition.

Continuous Time Parameter Process
The continuous time parameter queueing process is our main objective, which we target in the present section.The tools we are exploring are quite different from that used in our past experience (cf.Dshalalow [2]).Back then, we extended the process (Ω, F, (P x ) x=0,1,... , Z t ; t ≥ 0) from just being semi-regenerative to a two-(and more) variate process by using typical supplementary variable techniques, forming Kolmogorov's partial differential equations and Laplace transforms, then combining all of these with some semi-regenerative tools to yield a compact result.In this particular case, the past approach does not work since the additional information about arrival processes still does not make the two-variate process Markov, and any further efforts in this direction are cumbersome and counterproductive.So, we instead try to use a similar idea as that utilized in our recent paper [3] on semi-regenerative analysis directly applied to (Ω, F, (P x ) x=0,1,... , Z t ; t ≥ 0).Knowledge of invariant probability measures for embedded processes is crucial information needed for the upcoming analysis.We present some formal concepts of semi-regenerative processes pertinent to the models studied here.According to Section 3, is a semi-regenerative process relative to the sequence {T n } of stopping times.By the key convergence theorem for semi-regenerative processes, for each initial state x = 0, 1, . . ., the limiting probability exists if mµα > 1 and is given by the expression where {K ij (t)} is the semi-regenerative kernel of the process (Z t ) (cf. [2]) whose entries are defined as Observe that K ij (t) = P i {Z t = j T 1 > t} [1−A(x)], and thus Notice that and Let us now consider equations (4.2) for the index values j ≥ m.The following theorem states that, except for π m , the result for π j , j ≥ m, is similar to that of [6].
Theorem 4.1: The part of the limiting distribution π m , π m+1 , . . . of the queueing process Z t exists if mµα > 1, it is independent of any initial state, and is given by

Proof:
Case I: Let j > m.Then from (4.2), where and Inserting (4.8) into (4.6)we get Then, mµ , we have (using integration by parts for Stieltjes integrals as in Appendix, Theorem A.1) and Therefore, and the latter simplifies to or also in the form: (since where and Now we turn to ϕ(z).Substitution of (4.5) into (4.24) and Fubini's Theorem give where

.29)
The latter can be simplified to where and and U r is defined as in (3.43).
Proof: We use a method similar to that in Section 3. If With the usual combinatorics we arrive at m−1 i=r−1 where   In the following sections we will be concerned with a special case.

Special Case
Consider the special case of GI/M/ ma /∞, with a = 1 and p(x) = e −µx , which corresponds to the system GI/M/m/∞.Now, we have:  can hold while a 2 = ∞, and so condition (A.2) is exactly what we need and it is weaker.However, a 2 < ∞ is more tame.In most special cases, however, (A.2) holds true (i.e. if 1 − A(x) → 0 and x → ∞, the latter should go faster to zero than 1 x does ) .For instance, if A(x) is k -Erlang, then obviously vanishes when x → ∞, for any k = 1, 2, ..., and so will do any convex linear combination of Erlang distributions.