On the Transient Behavior of the  M / M /1 and  M / M /1/ M  Queues | Zendy

Xie Shisheng | Zendy; Knessl Charles | Zendy

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On the Transient Behavior of the M / M /1 and M / M /1/ M Queues

Author(s) -

Xie Shisheng,

Knessl Charles

Publication year - 1993

Publication title -

studies in applied mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.164

H-Index - 46

eISSN - 1467-9590

pISSN - 0022-2526

DOI - 10.1002/sapm1993883191

Subject(s) - queue , transient (computer programming) , burke's theorem , mathematics , representation (politics) , asymptotic analysis , mathematical analysis , mathematical physics , queueing theory , combinatorics , queue management system , computer science , statistics , fork–join queue , operating system , politics , political science , law , programming language

This paper gives a transient analysis of the classic M/M/ 1 and M/M/ 1/ K queues. Our results are asymptotic as time and queue length become simultaneously large for the infinite capacity queue, and as the system’s storage capacity K becomes large for the finite capacity queue. We give asymptotic expansions for p n ( t ), which is the probability that the system contains n customers at time t . We treat several cases of initial conditions and different traffic intensities. The results are based on (i) asymptotic expansion of an exact integral representation for p n ( t ) and (ii) applying the ray method to a scaled form of the forward Kolmogorov equation which describes the time evolution of p n ( t ).

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