The roles of coupling and the deviation matrix in determining the value of capacity in M/M/1/C queues

Abstract

In an M/M/1/C queue, customers are lost when they arrive to find C customers already present. Assuming that each arriving customer brings a certain amount of revenue, we are interested in calculating the value of an extra waiting place in terms of the expected amount of extra revenue that the queue will earn over a finite time horizon [0, t].There are different ways of approaching this problem. One involves the derivation of Markov renewal equations, conditioning on the first instance at which the state of the queue changes, a second involves expressing the value of capacity in terms of the entries of a transient analogue of the deviation matrix, discussed by Coolen-Schrijner and van Doorn in [6], and a third involves an elegant coupling argument.In this paper, we shall compare and contrast these approaches and, in particular, use the coupling analysis to explain why the value of an extra unit of capacity remains the same when the arrival and service rates are interchanged when the queue starts at full capacity.

The time-dependent expected reward and deviation matrix of a finite QBD process

“…see [4]. As t → ∞, D(t) converges to the deviation matrix D discussed in Coolen-Schrijner and van Doorn [7], which corresponds to the group inverse of −Q, and the expected lost revenue function has a linear asymptote, R(t) ∼ (πg)1 t + Dg.…”

“…We see that D = lim t→∞ D(t), where D(t) is the transient deviation matrix defined by (3). Properties of the transient deviation matrix are discussed in [4].…”

“…Our analysis of the expected reward in a finite QBD process is motivated by the problem of computing the expected amount of lost revenue in a MAP/PH/1/C queue over a finite time horizon [0, t], given its initial occupancy. Since MAPs and PH distributions are the most general matrix extensions of Poisson processes and exponential distributions, respectively, we can think of this problem as a matrix generalisation of the similar analysis for the M/M/C/C model considered in Chiera and Taylor [6] and the M/M/1/C model in Braunsteins, Hautphenne and Taylor [4]. In these queueing models, customers are lost when they arrive to find C customers already present.…”

Transient provisioning and performance evaluation for cloud computing platforms: A capacity value approach

“…Since obtaining analytic results for the transient distribution of loss network type systems is notoriously difficult, one often resorts to numerical inversion of Laplace transforms [3,24] or approximations [27]. In [13], its companion [12], and more recently [5] and [30], a useful alternative to the consideration of the transient distribution for queueing type models is proposed. In these papers the authors assume that tasks which fail to enter a loss system due to capacity constraints result in the system's manager incurring a predetermined amount of lost revenue.…”

Modelling complex stochastic systems: approaches to management and stability

“…Since obtaining analytic results for the transient distribution of loss network type systems is notoriously difficult, one often resorts to numerical inversion of Laplace transforms [128,129] or approximations [130]. In [115], its companion [116], and more recently [131] and also as discussed in the previous chapter, a useful alternative to the consideration of the transient distribution for queueing type models is proposed. In these papers the authors assume that tasks which fail to enter a loss system due to capacity constraints result The results in [116] and [115] are, however, derived using delicate manipulations of orthogonal polynomials, and it is difficult to use the same analytical techniques to generalize these findings to models which are more applicable to the distributed cloud computing setting.…”