| Abstract: | 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] Takâcs, L. 1960. Introduction to the Theory of Queues Oxford University Press. [Google Scholar] and Sharma and Gupta [13] Sharma, O.P. and Gupta, U.C. 1982. Transient Behaviour of an M/M/l/N Queue. Stoch. Process Appl., 13: 327–331. [Crossref] , [Google Scholar] 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. |
