A Storage Model with a Two-State Random Environment

Abstract

Motivated by queues with service interruptions, we consider an infinite-capacity storage model with a two-state random environment. The environment alternates between “up” and “down” states. In the down state, the content increases according to one stochastic process; in the up state, the content decreases according to another stochastic process. We describe the steady-state behavior of this system under assumptions on the component stochastic elements. For the special case of deterministic linear flow during the up and down states, we show that the steady-state content is directly related to the steady-state workload or virtual waiting time in an associated G/G/1 queue, thus supplementing the results of D. P. Gaver, Jr., and R. G. Miller, Jr. (1962), R. G. Miller, Jr. (1963) and H. Chen and D. D. Yao (1992).

Fluid queues with long-tailed activity period distributions

“…As might be expected: < 1 , < 1. Theorem 3.5 [31]. If < 1 and A 1 and S 1 have non-lattice distributions, then Z(t) !…”

“…Then the beginnings of activity and silence periods form a so-called alternating renewal process. This model is a specialization of the storage model introduced by Kella and Whitt [31]. If r 1 (t) r > 1, we shall mention it as an on/o source (the case r = 1 is trivial, since obviously the uid queue is constantly empty after some time).…”

“…The asymptotic behaviour of Z(t) has been investigated in [31], where the analysis relies on the arguments used for the virtual waiting time in the classical GI/G/1 queue (see Asmussen [3], Chapter VIII, 3). When dealing with the continuous time behaviour of the uid queue, we shall often use auxiliary parameters that naturally appear in formulas: the stationary probability of silence p := 1 1 + ; and the tra c intensity := ( + ) 1 + ; that is the long-run average amount of uid sent per time unit.…”

The workload-dependent MAP/PH/1 queue with infinite/finite workload capacity

“…Since a sample path of Y (t) can be obtained from the sample path of X (t) by deleting the time segments in which X (t) is increasing, solving for the steady-state distribution of X (t) will suffice as far as the steady-state distribution of Y (t) is concerned. Therefore, workload-dependent buffers can be analyzed using the paradigm of Markov fluid queues as in [32,9,11,12,31].…”

A queueing model with dependence between service and interarrival times

“…We present a detailed analysis of the joint distribution of workload and 'state of the arrival process', as well as of the waiting time distribution. We refer to Kella and \Vhitt [9] for a discussion of the equivalence relation between the workload process of the G I / G /1 queue and the buffer content process of a fluid model with linear flow and a two-state random environment. See [8] for an extension to the case of non-linear flow, and [3] for several equivalence relationships in the case of a three-state random environment.…”