Approximation of an M/M/s queue by the M/M/∞ one using the operator mathod

In this paper, we provide an approximate analysis of an M/M/s queue using the operator method (strong stability method). Indeed, we use this approach to study the stability of the M/M/∞ system (ideal system), when it is subject to a small perturbation in its structure (M/M/s is the resulting perturbed system). In other words, we are interested in the approximation of the characteristics of an M/M/s system by those of an M/M/∞ one. For this purpose, we first determine the approximation conditions of the characteristics of the perturbed system, and under these conditions we obtain the stability inequalities for the stationary distribution of the queue size. To evaluate the performance of the proposed method, we develop an algorithm which allows us to compute the various obtained theoretical results and which is executed on the considered systems in order to compare its output results with those of simulation.


Introduction
Multi-servers queues are analytically tractable queues of practical importance. However, the analytical results of such systems are only available in terms of Laplace transforms or/and generating functions which are often cumbersome and not useful in practice. To avoid and circumvent this problem, many authors use numerical and approximation methods to analyze such type of systems (see [5,6,7,8]). On the other hand, many situations are modeled by a multi-server queue, but for which we seek to have the average number of waiting customers to tend to zero. This situation coincides with an infinite-servers queue that has the non waiting property.
It is why we focus, in this work, on the strong stability method (see [1,11]) which allows us to make both qualitative and quantitative analysis helpful in understanding complicated models by more simpler ones for which an evaluation can be made. This method, also called "method of operators" can be used to investigate the ergodicity and stability of the stationary and non-stationary characteristics of the imbedded Markov chains (see [11]). In contrast to other methods, it supposes that the perturbations of the transition kernel are small with respect to some norms in the operators space. This stringent condition gives better stability estimates and enables us to find precise asymptotic expansions of the characteristics of the perturbed system.
The applicability of the strong stability method is well proved and documented in various fields and for different purposes. In particular, it has been applied to several proposals (see for example [2,3,4,12]). This paper aims to study the strong stability of an M/M/∞ system (ideal system) after a small perturbation of its structure (the M/M/s is the resulting perturbed system). We first clarify the conditions for such an approximation, then we provide an upper bound for the norm of the difference between the two stationary distributions of the considered systems. Note that there is a first attempt to study this particular case of approximation of the M/M/s queue by the M/M/∞ one, using the theory of Markov chains and the special norm L 1 (see [9]). The authors of this latter work had produced two important results. The first one is summarized in the upper bound of the absolute difference (L 1 norm) between the stationary probabilities of the M/M/s and M/M/∞ systems, and in the second point the authors have proved that this difference tends to zero when the number of servers tends to infinity. Unlike [9], we use here the general weight norm ( . v ) instead of the norm L 1 to approach the M/M/s system by the M/M/∞ one. Moreover, we provide the domain within which this approximation is valid.

Description of M/M/s and M/M/∞ models
Let us consider an M/M/s system with s servers, where inter-arrival times are independently distributed with an exponential distribution E λ (t) and mean inter-arrival time 1/λ, and service times are distributed with E µ (t) (exponential with parameterµ). Let X k be the number of customers in the system just prior to the arrival of the k th customer. Therefore, X = (X k ; k = 0, 1, . . .) is an homogeneous Markov chain with a state space N = {0, 1, 2, . . .} and with a transition operator P = (P ij ) i,j≥0 where: Consider also an M/M/∞ system, with the same distributions of arrivals and service times as in the previous system, and with no waiting room. LetX k be the number of customers in the system just prior to the arrival of the k th customer. Therefore,X = (X k ; k = 0, 1, . . .) is an homogeneous Markov chain with a state space N = {0, 1, 2, . . .} and with a transition operatorP = (P ij ) i,j≥0 where: Let π andπ be, respectively, the stationary distributions of the M/M/s and M/M/∞ systems.
Introduce on M the υ-norm of the form: where υ(k) = β k , for all k ∈ N and β > 1 is a real parameter. This norm induces in the space N the norm Moreover, for all ν ∈ M and f ∈ N , the symbols νf and f •ν denote respectively the summation and the kernel defined as below Let us consider B, the space of linear operators, with the norm

Strong stability criteria
For more details on the strong stability method, see [1,10,11]. Let us give the definition of the strong stability (qualitative aspect) for an homogeneous Markov chain in the phase state (N, B(N)) with respect to the υ-norm. Here B(N) is the σ-algebra generated by the singletons {j}.
Definition 1 (see [1,11]) A Markov chain X with a transition kernel P and an invariant measure π is said to be strongly υ-stable with respect to the norm . υ if P υ < ∞ and each stochastic kernel Q on the space (N, B(N)) in some neighborhood {Q : Q − P < } has a unique invariant measure µ = µ(Q) and π − µ υ → 0 as Q − P υ → 0 4. Strong stability in the M/M/∞ system 4.1. Strong stability conditions Proofs of the results provided in this section, are based on theoretical results (theorems and corollaries) given in [1,10].
and under the condition: we have Where α = λ/µ.

Numerical application
To be able to put into practice the previous theoretical results concerning the approximation of the M/M/s system by the M/M/∞ one, we have developed the following algorithm:

Algorithm
Step 1. Introduce input parameters: inter-arrivals mean rate λ, number of servers s and service mean rate µ.
Step 2. Verify the existence of β 0 if µ > λ then the system is stable and goto step 3 else disp 'the system is not stable' and goto step 7.
For numerical applications, we consider the two following situations:         see table 2). This highlights the role of the load α of the system in its stability. • The conditions and the stability bound of the system depend exponentially on the parameters of the system. Indeed, the error (B β ) increases exponentially with the increase of the load of the system α (see figure 5), where it passes from the stability state to the instability state starting from the value α = 0.4250 (see table 3). This means that the stability of the system is strictly related not only to the number of its servers but also to its load. • According to figures 2, 4 and 6, we notice that the simulation results are always lower than the algorithmic ones. This confirms that the bound (B β ) is an upper bound for the deviation π −π υ .

Conclusion
In this work, we applied for the first time the strong stability method on the M/M/∞ queue which is subject to a small perturbation of its structure. The application of the method in question, allows us to determine the stability conditions of the M/M/∞ system and to obtain stability bounds for the stationary characteristics of the M/M/s system by those of the M/M/∞ one. To validate and to illustrate the manner in which the theoretical results can be exploited in practice, we have presented numerical examples based on simulation studies. We may extend this analysis to the approximation case of the GI/M/s system by the GI/M/∞ one.