The impact of the service discipline on delay asymptotics

S.C. Borst, O.J. Boxma, R. Núñez-Queija, A.P. Zwart
2003 Performance evaluation (Print)  
This paper surveys the M/G/1 queue with regularly varying service requirement distribution. It studies the effect of the service discipline on the tail behavior of the waiting-time and/or sojourn-time distribution, demonstrating that different disciplines lead to quite different tail behavior. The orientation of the paper is methodological: We outline four different methods for determining tail behavior, illustrating them for service disciplines like FCFS, Processor Sharing and LCFS. S.C. Borst
more » ... et al. / Performance Evaluation 54 (2003) P{T = k} ∼ αk −(α+1) L(k), where 1 < α < 2 and L(·) is a slowly varying function (see Definition 2.1). Each session brings in work at unit rate while it is active. Hence, the work brought in by each arrival is regularly varying and, because 1 < α < 2, the arrival process of work is long-range dependent, but E{T } < ∞. Anantharam shows that, in the steady-state case, for any stationary Non-Preemptive service policy, the sojourn-time of a typical session must stochastically dominate a regularly varying random variable having infinite mean. Non-preemption means that once service on a session has begun, it is continued until all the work associated with it has been completed. Anantharam does not make any assumptions as to whether the service policy is work-conserving, or whether the length of a session is known at the time of arrival. In contrast, Anantharam further shows that there also exist causal stationary preemptive policies, which do not need information about the session durations at the time of their arrival, for which the sojourn-time of a session is stochastically dominated by a regularly varying random variable with finite mean. The results of Anantharam raise several questions, like (i) are there (preemptive) service disciplines for which the tail of the sojourn-time distribution is not heavier than the tail of the service requirement distribution, and (ii) what is the effect of various well-known scheduling disciplines on the tail behavior of the waiting-time and/or sojourn-time distribution? A related issue arises when there are several classes of customers, which may be treated in different ways by the server (e.g., using fixed priorities, or according to a polling discipline). Then it is important to understand under what conditions, or to what extent, the tail behavior of the service requirements of one class affects the performance of other classes. The above issues have recently been investigated by the present authors and some of their colleagues. This paper summarizes the results. We focus on the classical M/G/1 queue and its multi-class generalizations (although some of the recently obtained results allow a general renewal arrival process, or a fluid input). The orientation of the paper is methodological. After introducing the model and reviewing the main results for various basic disciplines in Section 2, we discuss four different methods for obtaining the tail behavior of waiting-time and/or sojourn-time distributions for M/G/1-type queues with regularly varying service requirement distribution(s): (i) an analytical one, which relies on Tauberian theorems relating the tail behavior of a probability distribution to the behavior of its Laplace-Stieltjes transform near the origin; (ii) a probabilistic one, which exploits a Markov-type inequality, relating an extremely large sojourn (or waiting) time to a single extremely large service requirement; (iii) a probabilistic one, which is based on sample-path arguments which lead to lower and upper bounds for tail probabilities; (iv) a probabilistic one, which is based on explicit (random-sum) representations of the waiting-time distribution, which are applicable to the larger class of subexponential distributions. These four approaches are described in Sections 3-6, respectively. Sections 3, 5 and 6 also discuss the multi-class case. Concluding remarks are given in Section 7. The present paper is an extended version of [16] . In the present version, minor changes have been made in Sections 3 and 4, Section 5 is significantly improved at several points, and Section 6 is new. Model description and main results In this section, we formally describe the model, introduce some concepts and notation, and give an overview of the main results. As mentioned earlier, we focus on the M/G/1 queue. In this system, customers arrive according to a Poisson process, with rate λ, at a single server who works at unit rate. Their service requirements
doi:10.1016/s0166-5316(03)00071-3 fatcat:nuto7ak3unc4bgc2q6uqbxdwva