The transient solution of time-dependent M/M/1 queues

The transient behavior of time-dependent M/M/1 queues is studied. The boundary probability function pi /sub 0/(t), which is the probability that the queue is empty at time t, is shown with analyticity arguments to satisfy a Volterra-type integral equation. The boundary integral equation is derived, and a numerical algorithm is used to solve the integral equation and to find the expected queue size from pi /sub 0/(t). The approach can be applied to many other types of time-dependent queues. Examples are given. >

[1]  M. R. Taaffe,et al.  Approximating nonstationaryPh(t)/M(t)/s/c queueing systems , 1987 .

[2]  Ward Whitt,et al.  Transient behavior of the M/M/l queue: Starting at the origin , 1987, Queueing Syst. Theory Appl..

[3]  G. E. H. Reuter.  Denumerable Markov Processes (II) , 1959 .

[4]  William A. Massey,et al.  Asymptotic Analysis of the Time Dependent M/M/1 Queue , 1985, Math. Oper. Res..

[5]  S. McKee,et al.  Variable step size predictor-corrector schemes for second kind Volterra integral equations , 1985 .

[6]  G. F. Newell.  Queues with time-dependent arrival rates I—the transition through saturation , 1968 .

[7]  M. S. Bartlett,et al.  On the differential equations for the transition probabilities of Markov processes with enumerably many states , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  Kenneth Lloyd Rider.  A Simple Approximation to the Average Queue Size in the Time-Dependent M/M/1 Queue , 1976, JACM.

[9]  A. Clarke,et al.  A Waiting Line Process of Markov Type , 1956 .

[10]  W. Grassmann.  Transient solutions in Markovian queues : An algorithm for finding them and determining their waiting-time distributions , 1977 .

[11]  Ernst Hairer,et al.  FAST NUMERICAL SOLUTION OF NONLINEAR VOLTERRA CONVOLUTION EQUATIONS , 1985 .

[12]  G. Reuter,et al.  Denumerable Markov processes and the associated contraction semigroups onl , 1957 .

[13]  Shmuel S. Oren,et al.  A Closure Approximation for the Nonstationary M/M/s Queue , 1979 .