Comparative analysis of a randomized N-policy queue: An improved maximum entropy method

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Abstract

We analyze a single removable and unreliable server in an M/G/1 queueing system operating under the 〈p, N〉-policy. As soon as the system size is greater than N, turn the server on with probability p and leave the server off with probability (1  p). All arriving customers demand the first essential service, where only some of them demand the second optional service. He needs a startup time before providing first essential service until there are no customers in the system. The server is subject to break down according to a Poisson process and his repair time obeys a general distribution. In this queueing system, the steady-state probabilities cannot be derived explicitly. Thus, we employ an improved maximum entropy method with several well-known constraints to estimate the probability distributions of system size and the expected waiting time in the system. By a comparative analysis between the exact and approximate results, we may demonstrate that the improved maximum entropy method is accurate enough for practical purpose, and it is a useful method for solving complex queueing systems.

Introduction

In this paper, we consider an unreliable server in an M/G/1 queue operating under the 〈p, N〉-policy with a second optional service (here abbreviated as SOS) and general startup times. An unreliable server means that the server is typically subject to unpredictable breakdowns. We elaborate an information theoretic technique based on the principle of maximum entropy to give an alternative solution for deriving probability distributions in this queueing model. We call that the policy is a 〈p, N〉-policy if it prescribes the following conditions: (i) turn the server off when the system is empty, (ii) turn the server on if there are N(N  1) or more customers are present, (iii) if the server is turned off and the number of customers in the system reaches N, turn the server on with probability p and leave the server off with probability (1  p), and (iv) do not turn the server at other epochs. If the server finds at least N customers present in the system, it starts to provide first essential service (here abbreviated as FES) for the waiting customers whenever he completes its startup. In other words, the 〈p, N〉-policy is to control the server randomly at the arrival epoch of the Nth customer finds that the server is idle. If the probability p is one, then we have N-policy introduced by Yadin and Naor (1963). In case p = 0, we have the (N + 1)-policy. An M/G/1 queue involving the randomized server control problem has been treated by Feinberg and Kim (1996). They considered either 〈p, N〉- or 〈N, p〉-policy M/G/1 queue with a removable sever at first and performed the optimal control policy is of the randomized form. Subsequently, Kim and Moon (2006) considered the system with the 〈p, T〉-policy, exploit its properties and found the optimal values of T and p for a constrained problem. Lately, Ke, Ko, and Sheu (2008) utilized bootstrap methods to investigate the estimation of the expected busy period of an M/G/1 queueing system under 〈p, N〉-policy.

One of the most significant regions of queueing problem is the control of queue, and have studied extensive by many researchers. Yadin and Naor (1963) first introduced the concept of an N-policy which turns the server on whenever N (N  1) or more customers are present, turns the server off only when the system is empty. The server startup corresponds to the preparatory work of the server before starting the service. In some actual situations, the server often needs a startup time before providing service. Exact steady-state solutions of the N policy M/M/1 queue with exponential startup times were first derived by Baker (1973). Borthakur, Medhi, and Gohain (1987) extended Baker’s model to general startup times. Wang (2003) developed the exact steady-state solutions of the N policy M/M/1 queue with server breakdowns and exponential startup times. The N-policy M/G/1 queue with startup times was investigated by several authors such as Medhi and Templeton, 1992, Takagi, 1993, Lee and Park, 1997, etc. Ke (2003) analyzed the N policy M/G/1 queueing system with server vacations, startup and breakdowns. He developed the probability generating function of the queue size when the server begins performing startup and also derived important system characteristics. Recently, Wang, Wang, and Pearn (2007) focused mainly on performing a sensitivity analysis for the N-policy with server breakdowns and general startup times.

In many real service systems, one encounters numerous examples of the queueing situation where all arrivals require the main service and only some may require the subsidiary service provided by the server. Madan (2000) was the first to study an M/G/1 queue with SOS in which the first essential service time obeys a general distribution but second optional service time follows an exponential distribution. He also cited some important examples in daily life. Medhi (2002) extended Madan’s model (Madan, 2000) that the second optional service time follows a general distribution. Al-Jararha and Madan (2003) generalized Madan’s work in the sense that they assumed that both first essential service time and second optional service time are general with different distribution functions. Based on the supplementary variable technique, Wang (2004) studied the reliability behavior in an M/G/1 queue with SOS and an unreliable server. Recently, Wang and Zhao (2007) considered a discrete-time Geo/G/1 retrial queue with an unreliable server and SOS. Some performance measures of the system in steady state and explicit formulae for the stationary distribution are developed in their work.

In a stochastic context, little is known analytically about the behaviors of queue length distributions of a randomized server control queueing system. When exact methods of solution are not known, we frequently make use of numerical solution methods. One elegant approach for this is given by an information theoretic technique, which based on the principle of maximum entropy, to provide a self-contained method of inference for obtaining an unknown and unique probability distribution. In other word, this method is applied to estimate probability distributions, which consists of maximizing entropy function subject to the available mean constraints. El-Affendi and Kouvatsos (1983) presented the maximum entropy formalism to analyze the M/G/1 and G/M/1 queues. Based on the maximum entropy principle, Artalejo and Lopez-Herrero (2004) investigated the probability density function of busy period under some controllable M/G/1 queueing models. Wang, Wang, and Pearn (2005) used maximum entropy analysis to study the N policy M/G/1 queueing system with server breakdowns and general startup times. Recently, Ke and Lin (2006) studied the M[x]/G/1 queueing system with an unreliable server and delaying vacations. They derived the approximate steady-state probability distribution of the queue length as well. To the best of our knowledge, that there has been no research that investigates a randomized controllable queueing system with SOS and startup times by the maximum entropy principle. Our work is motivated by such works and employ maximum entropy method to estimate the queue length distribution for the 〈p, N〉-policy M/G/1 queue with server breakdowns, SOS and startup times.

The purpose of this paper is fourfold. Firstly, we develop some exact and important system performance measures for the 〈p, N〉-policy M/G/1 queue with server breakdowns, SOS and startup times. Secondly, we construct an improved maximum entropy function for this queueing system. Thirdly, the improved maximum entropy solutions are developed through the Lagrange’s method. Thirdly, we obtain the approximate expected waiting time in the system and the exact expected waiting time in the system. Finally, we perform a comparative analysis between approximate results obtained through the improved maximum entropy method and exact results obtained from the convex combination property.

Section snippets

The mathematical model

In this paper, we consider the 〈p, N〉 M/G/1 queue with the following specifications. It is assumed that customers arrive according to a Poisson process with rate λ. Arriving customers form a single waiting line at a server based on the order of their arrivals; that is, in a first-come, first-served (FCFS) discipline. A single server is required to serve all arriving customers for the first essential service (FES), denoted by S1. As soon as FES of a customer is completed, a customer may leave the

System performance measures

Let H1 and H2 be a random variable representing the completion time of FES and SOS, respectively. The completion time of a customer includes both the service time of a customer and the repair time of a server. Using the known results of Wang and Ke (2002), we get the first two moments of the completion time distribution for the first essential channel and second optional channel:E[Hi]=μSi1+αiμRi,i=1,2,EHi2=1+αiμRi2ESi2+αiμSiERi2,i=1,2.

We denote by IN, UN, BN and DN, idle, startup, busy,

Improved maximum entropy results

Exact probability distributions of the 〈p, N〉-policy M/(G, G), (G, G), G/1 queue have not been found. Therefore, we employ the improved maximum entropy principle to estimate probability distributions of the number of customers given several known results. In this section, we will develop the improved maximum entropy solutions for the steady-state probabilities of the 〈p, N〉-policy M/(G, G), (G, G), G/1 queue.

The exact and approximate expected waiting time in the system

In this section, we first derive the exact expected waiting time in the system by using Little’s formula. Through the maximum entropy principle, the approximate formulae of the expected waiting time in the system for the 〈p, N〉-policy M/(G, G), (G, G), G/1 queue is developed.

Comparative analysis between exact and approximate results

This section aims to examine the accuracy of the approximate results based on the improved maximum entropy principle. We provide numerical comparisons between the exact results and the approximate results, including various service time, startup time and repair time distribution functions. There are three subsections in the following:

  • (1)

    Comparative analysis for the 〈p, N〉-policy M/(M, E2), (M, D),M/1 queue.

  • (2)

    Comparative analysis for the 〈p, N〉-policy M/(M, D), (E2, E3),D/1 queue.

  • (3)

    Comparative analysis for

Conclusion

In this paper, we developed some important system performance measures for the 〈p, N〉-policy M/(G, G), (G, G), G/1 queue. An elegant approach, the maximum entropy principle, is used to derive the approximate formulae for the steady-state probability distributions of the queue length. Our numerical investigations show that it is feasible to use the probability of various server states and the expected number of customers in the system when the server is not idle. The numerical results also indicate

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