Sofiane Ouazine1 and Karim Abbas2
1 Department of Mathematics
University of Bejaia, Campus of Targua Ouzemour, Algeria (e-mail: wazinesofi@gmail.com)
2 LAMOS Laboratory,
University of Bejaia, Campus of Targua Ouzemour, Algeria (e-mail: kabbas.dz@gmail.com)
Abstract. In this paper we discuss the applicability of the Taylor series approach to the numerical analysis of the GI/M/1 queue with negative customers. In other words, we use the Taylor series expansions to examine the robustness of the GI/M/1 (FIFO,
∞) queueing model having RCH (Removal of Customer at the Head) to perturbations in the negative customers process (the occurrence rate of RCH). We analyze numeri-cally the sensitivity of the entries of the stationary distribution vector of the GI/M/1 queue with negative customers to those perturbations, where we exhibit these entries as polynomial functions of the occurrence rate of RCH parameter of the considered model. Numerical examples are sketched out to illustrate the accuracy of our ap-proach.
Keywords: Taylor series expansion, Sensitivity analysis, GI/M/1 queue with nega-tive customers, Numerical methods, Performance measures.
1 Introduction
Recently there has been a rapid increase in the literature on queueing systems with negative arrivals. Queues with negative arrivals, called G-queues, were first introduced by Gelenbe [5]. When a negative customer arrives, it imme-diately removes an ordinary (positive) customer if present. Negative arrivals have been interpreted as inhibiter and synchronization signals in neural and high speed communication network. For example, we can use negative arrivals to describe the signals, which are caused by the client, cancel some proceeding.
There is a lot of research on queueing system with negative arrivals. But most of these contributions considered continuous-time queueing model: Boucherie and Boxma [6], Jain and Sigman [8], Bayer and Boxma [2], Harrison and Pitel [9] all of them investigated the same M/G/1 model but with the different killing strategies for negative customers; Harrison, Patel and Pitel [10] considered the M/M/1 G-queues with breakdowns and repair; Yang [11] considered GI/M/1 model by using embedded Makov chain method.
3rdSMTDA Conference Proceedings, 11-14 June 2014, Lisbon Portugal C. H. Skiadas (Ed)
c 2014 ISAST
In this paper we investigate the GI/M/1/N with Poisson negative arrivals to test the robustness of the model to perturbation in the negative customers process (the occurrence rate of RCH). In deed, we use the Taylor series expan-sions to examine the robustness of the GI/M/1/N queue to perturbations in the arrival process. Specifically, we analyse numerically the sensitivity of the entries of the stationary distribution vector of the GI/M/1/N queue to that perturbations, where we exhibit these entries as polynomial functions of the occurrence rate of RCH.
The remainder of this paper is organized as follows. In Section 2, we intro-duce the necessary notations for analyzing of the considered queueing model, and present closed-form expressions for the sensitivity of the stationary distri-bution to model parameter as a function of the deviation matrix. In Section 3, we outline the numerical framework to compute the relative absolute error in computing the stationary distribution. Concluding remarks are provided in Section 4.
2 Queueing Model Analysis
We investigate the GI/M/1/N queue with negative customers, whereN is the capacity of the system including the one who is in service. Assume that cus-tomer arrivals occur at discrete-time instantsτk, whereτ0= 0, customers arrive at the system according to a renewal process with interarrival time distribu-tionG(t) and mean 1/λ. The service timeTs of each server is assumed to be distributed exponentially with service rateµ. Its density function is given by
s(t) =µe−µt, t≥0.
Additionally, we assume that there is another kind of customers, namely RCH, arriving in the system according to an independent Poisson process of pa-rameter h. Let Lk denote the number of customers left in the system im-mediately after the kth departing customer. A sequence of random variables Lk;k= 1,2, . . . , N constitutes a Markov chain. Its transition probabilities the Laplace transformation corresponding to pdfs f(i.edG(t) =f(t)dt) of the interarrival process.
Letπdenote the stationary distribution of the Markov chainLk. We define the traffic intensity ρ=(arrival rate)/(service rate) = λ/µ. It can be shown
that the Markov chainLk is positive recurrent for allρ. In this paper, we con-sider the stationary distributionπas a mapping of some real-valued parameter θ, in notationπθ. For example,θ may denote the occurrence rate of RCH pa-rameter of the model. We are interested in obtaining higher-order sensitivity of stationary distribution with respect to parameterθ. In the sequel we derive formulas for the higher order derivatives of πθ with respect to θ. Then, by using these formulas we obtain a Taylor series expansions inθforπθ+∆, where its coefficients are expressed in closed form as functions of thedeviation matrix (denoted byDθ) associated to the Markov chainLk. It is well know that ifPθ
is irreductible, then (I−Pθ+Πθ) is invertible, where Πθ =e πθ. Then, the matrixDθ= (I−Pθ+Πθ)−1−Πθexists and it is called the deviation matrix.
The deviation matrix can be obtained in explicit form by:
Dθ=
∞
X
n=0
(Pθn−Πθ),
=
∞
X
n=0
(Pθ−Πθ)n−Πθ,
= (I−Pθ+Πθ)−1−Πθ.
In the following theorem we give the higher-order derivatives of the stationary distributionπθwith respect toθin terms of the deviation matrixDθ, which is a key result used in the framework proposed subsequently.
Theorem 1. [7]Letθ∈Θ and letΘ0⊂Θ, withΘ⊂Rbe a closed interval withθan interior point such that the Markov chain is ergodic onΘ0. Provided that the entries of the transition matrix Pθ are n-times differentiable with respect to θ, let
Kθ(n) = X
1≤m≤n;
1≤lk ≤n;
l1+· · ·+lm=n n!
l1!· · ·lm!
m
Y
k=1
Pθ(lk)Dθ
..
Then, it holds that
πθ(n) = πθKθ(n), (1)
wherePθ(k)(respectivelyπ(k)θ ) is the matrix (respectively vector) of the elemen-twisekth order derivative ofPθ (respectivelyπθ) with respect to parameter θ.
In the following, we propose a numerical approach to compute the station-ary distributionπθ for some parameter valueθ, and we demonstrate how this stationary distribution can be evaluated for the case where the control parame-terθis changed in some interval. In other words, we will compute the function π(θ+∆) on some∆-interval. More specifically, we will approximately compute π(θ+∆) by an polynomial in∆. To achieve this we will use the Taylor series
expansion approach established in [7]. Under some mild conditions it holds that πθ+∆ can be developed into a Taylor series of the following form:
πθ+∆=
k
X
n=0
∆n
n! πθ(n), (2)
whereπ(n)θ denotes then-th order derivative ofπθwith respect toθ(see formula (1)).
We call
Hθ(k, ∆) =
k
X
n=0
∆n
n! πθ(n) (3)
thek-th order Taylor approximation ofπθ+∆at θ.
Under the conditions put forward in Theorem 1 it holds fork < nthat:
πθ(k+1) =
k
X
m=0
k+ 1 m
πθ(m)Pθ(k+1−m)Dθ. (4)
An explicit representation of the lower derivatives ofπθ is given by [1]:
πθ(1)=πθPθ(1)Dθ (5) and
πθ(2)=πθPθ(2)Dθ+ 2πθ(Pθ(1)Dθ)2. (6) Elaborating on the recursive formula for higher order derivatives (4), the second order derivative can be written as:
πθ(2)=πθPθ(2)Dθ+ 2πθ(1)Pθ(1)Dθ. (7) In the same vein, we obtain for the third order derivative:
πθ(3)=πθPθ(3)Dθ+ 3π(2)θ Pθ(1)Dθ+ 3πθ(1)Pθ(2)Dθ. (8)
3 Numerical Application
In this section, we apply the numerical approach based on the Taylor series ex-pansions introduced above to the GI/M/1/N queue with negative customers, where we consider the model with perturbed the occurrence rate of RCH pa-rameter. In this case, we estimate numerically the sensitivity of the stationary distribution of the queueing model with respect to the perturbation.
LetΘ= (a, b)⊂R, for 0< a < b <∞.
(H) For 0≤j≤N it holds thataj isn-times differentiable with respect to honΘ.
Under(H)it holds that the firstnderivatives ofP exists. LetP(k)denote thekth order derivative ofP with respect toh, then it holds that
P(k)(i, j) = d(k)
Consider the M/M/1/N queue with service rateµ and exponential inter-arrival time with rate λ. First, we present the numerical results obtained by applying our approach to this case. Therefore, we setµ= 2, λ= 1 . For the implementation of our algorithm in MATLAB, we require a finite version of our queueing model. Figures 1, 2 and 3 depict the relative error on the stationary distributions πθ(k)(i) for 0 ≤ i ≤ N and k = 1,2,3, of the M/M/1/N queue versus the perturbation parameter ∆ ∈ [−δ, δ], where δ = 0.1. As expected, the relative error on the stationary distributions decreases as the perturbation parameterhdecreases.
Fig. 1.The relative error in computingπ1+∆by Taylor series of 1st order.
series coefficients are given in terms of the deviation matrix corresponding to the embedded Markov chain. We have presented some numerical examples
Fig. 2.The relative error in computingπ1+∆by Taylor series of 2nd order.
Fig. 3.The relative error in computingπ1+∆by Taylor series of 3rd order.
that illustrate our numerical approach. In fact, the convergence rate of the Taylor series is such that already a Taylor polynomial of degree 2 or 3 yields good numerical results. As part of future work, we will further investigate the multi-server queues with vacations. We will also further provide a simplified and easily computable expression bounding the remainder of the Taylor series and, thereby provide an algorithmic way of deciding which order of the Tay-lor polynomial is su?cient to achieve a desired precision of the approximation Abbas, Heidergott and Aissani (2013).
4 Conclusion
This paper has developed a numerical, method to analyze the effect of the perturbation of the negative customers process in the performance measures of the queuing model considered (Stationary distribution), our numerical in-vestigation are based on the Taylor series expansion; see [7], where the Taylor series coefficients are given in terms of the deviation matrix corresponding to the embedded Markov chain. Therefore, we have presented different examples that illustrate our numerical approach, and as illustrated by the numerical ex-amples the convergence rate of the Taylor series is such that already a Taylor polynomial of degree 2 or 3 yields good numerical results, we will further in-vestigate the multi-server queues with vacations. We will also further provide a simplified and easily computable expression bounding the remainder of the Taylor series and, thereby provide an algorithmic way of deciding which
or-der of the Taylor polynomial is sufficient to achieve a desired precision of the approximation [1].
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