A single-server finite-capacity queueing system with Markov flow and discrete-time service

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.

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Analysis of a Discrete-Time Queueing Model with Disasters

Mathematics ◽

10.3390/math9243283 ◽

2021 ◽

Vol 9 (24) ◽

pp. 3283

Author(s):

Mustafa Demircioglu ◽

Herwig Bruneel ◽

Sabine Wittevrongel

Keyword(s):

Discrete Time ◽

Queueing System ◽

Time Slot ◽

Probability Distributions ◽

Single Server ◽

Bernoulli Process ◽

Tail Probabilities ◽

Mean Values ◽

The Mean ◽

The Impact

Queueing models with disasters can be used to evaluate the impact of a breakdown or a system reset in a service facility. In this paper, we consider a discrete-time single-server queueing system with general independent arrivals and general independent service times and we study the effect of the occurrence of disasters on the queueing behavior. Disasters occur independently from time slot to time slot according to a Bernoulli process and result in the simultaneous removal of all customers from the queueing system. General probability distributions are allowed for both the number of customer arrivals during a slot and the length of the service time of a customer (expressed in slots). Using a two-dimensional Markovian state description of the system, we obtain expressions for the probability, generating functions, the mean values, variances and tail probabilities of both the system content and the sojourn time of an arbitrary customer under a first-come-first-served policy. The customer loss probability due to a disaster occurrence is derived as well. Some numerical illustrations are given.

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A solvable model for a finite-capacity queueing system

Journal of Applied Probability ◽

10.1017/s0021900200108137 ◽

1985 ◽

Vol 22 (04) ◽

pp. 903-911 ◽

Cited By ~ 5

Author(s):

V. Giorno ◽

C. Negri ◽

A. G. Nobile

Keyword(s):

Queueing System ◽

Waiting Times ◽

Queueing Systems ◽

Solvable Model ◽

Probability Density Functions ◽

Single Server ◽

Finite Capacity ◽

System Service ◽

Transient Probabilities ◽

Number Of Customers

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.

Download Full-text