A solvable model for a finite-capacity queueing system

Abstract

Single–server–single-queue–FIFO-discipline queueing systems are considered in which at most a finite number of customers N can be present in the system. Service and arrival rates are taken to be dependent upon that state of the system. Interarrival intervals, service intervals, waiting times and busy periods are studied, and the results obtained are used to investigate the features of a special queueing model characterized by parameters (λ (Ν –n), μn). This model retains the qualitative features of the C-model proposed by Conolly [2] and Chan and Conolly [1]. However, quite unlike the latter, it also leads to closed-form expressions for the transient probabilities, the interarrival and service probability density functions and their moments, as well as the effective interarrival and service densities and their moments. Finally, some computational results are given to compare the model discussed in this paper with the C-model.

Time-Inhomogeneous Finite Birth Processes with Applications in Epidemic Models

“…, N, by setting µ(t) = 0. The Prendiville process is considered in studies by Giorno et al [26], Zheng [27], Giorno and Nobile [28], and Usov et al [29]. Proposition 4.…”

Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration

“…We consider the NHBI process N(t), with t 0 = 0, characterized by λ n (t) = λ n + ν(t), with λ > 0, subject to periodic immigration phenomena that occur with intensity function (25). In Figure 7, the transition probabilities p j,n (t|0), given in (42) and in (43), for the NHBI process with constant birth rate and periodic immigration intensity function (25) are plotted as function of t for j = 0 and n = 0, 1, 2 for some choices of parameters.…”

“…The Prendiville process has been extensively apply in biology, ecology and epidemiology to describe biological population growth in the limited environment (cf., for instance, Dharmaraja [17], Zheng [41] and Giorno et al [42]). In Giorno et al [43] the time-homogeneous Prendiville process has been used to analyze an adaptive queuing system model with finite capacity, in which the customers are discouraged to join the queue when the queue size is large and, at the same time, the server accelerates the service. For j, n ∈ {0, 1, .…”

A solvable model for a finite-capacity queueing system

“…The above alternative argument for deriving an(t) works only for the specific (An, J..tn) used in the binomial model considered in [3]. For general rates an(t) is from (2.2) of [3] the p.d.I, of a finite-mixture of generalised Erlangs.…”

“…For general rates an(t) is from (2.2) of [3] the p.d.I, of a finite-mixture of generalised Erlangs. an (t) can thus be expressed as a general exponential mixture density.…”