The equivalence of the transient behaviour formulae for the single server queue

• Published in: Stochastic analysis and applications Vol. 20; no. 2; pp. 415 - 426
• Main Author: Tarabia, A. M. K.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Taylor & Francis Group 22.05.2002; Taylor & Francis
• Subjects: AMS Subject Classifications: Primary: 60K20; Secondary 90B22; Exact sciences and technology; Maclaurin's expansion; Mathematics; Probability and statistics; Probability theory and stochastic processes; Recurrence relations; Sciences and techniques of general use; Single server queue; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Transient behaviour; Sharma formula; Takacs formula; Recurrence relation; Chebyshev polynomial; Probability; Computing; Stochastic process; Single server queue; Transients; Queueing system; Gupta formula; Behavior; Queue
• ISSN: 0736-2994; 1532-9356
• DOI: 10.1081/SAP-120003442

It is indeed obvious to expect that the different results obtained for some problem are equal, but it needs to established. For the M/M/1/N queue, using a simple algebraic approach we will prove that the results obtained by Takâcs [17] and Sharma and Gupta [13] are equal. Furthermore, a direct proof to the equivalence between all formulae of the M/M/l/∞ queue is established. At the end of this paper, we will show that the time-dependent state probabilities for M/M/l/N queue can be written in series form; its coefficients satisfy simple recurrence relations which would allow for the rapid and efficient evaluation of the state probabilities. Moreover, a brief comparison of our technique, Sharma and Gupta's formula and Takâcs result is also given, for the CPU time computing the state probabilities.