Abstract
We consider the M(t)/M(t)/m/m queue, where the arrival rate λ(t) and service rate μ(t) are arbitrary (smooth) functions of time. Letting p n (t) be the probability that n servers are occupied at time t (0≤ n≤ m, t > 0), we study this distribution asymptotically, for m→∞ with a comparably large arrival rate λ(t) = O(m) (with μ(t) = O(1)). We use singular perturbation techniques to solve the forward equation for p n (t) asymptotically. Particular attention is paid to computing the mean number of occupied servers and the blocking probability p m (t). The analysis involves several different space-time ranges, as well as different initial conditions (we assume that at t = 0 exactly n0 servers are occupied, 0≤ n0≤ m). Numerical studies back up the asymptotic analysis.
Similar content being viewed by others
References
M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
N. Bleistein and R.A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986).
R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume II (Interscience, New York, 1962).
S. Eick, W.A. Massey and W. Whitt, The physics of the M(t)/G/∞ queue, Operations Research 41 (1993) 400–408.
S. Eick, W.A. Massey and W. Whitt, Infinite-server queues with sinusoidal arrival rates, Management Science 39 (1993) 241–252.
A.K. Erlang, Solutions of some problems in the theory of probabilities of significance in automatic telephone exchanges, The Post Office Electrical Engineers’ Journal 10 (1918) 189–197.
D.L. Jagerman, Some properties of the Erlang loss function, Bell System Technical Journal 53 (1974) 525–551.
D.L. Jagerman, Nonstationary blocking in telephone traffic, Bell System Technical Journal 54 (1975) 625–661.
J.B. Keller, Time-dependent queues, SIAM Review 24 (1982) 401–412.
L. Kleinrock, Queueing Systems, Volume I: Theory (John Wiley, New York, 1975).
C. Knessl, On the transient behavior of the Erlang loss model, Comm. Statist. Stochastic Models 6 (1990) 749–776.
C. Knessl, Exact and asymptotic solutions to a PDE that arises in time-dependent queues, Advances in Applied Probability 32 (2000) 256–283.
C. Knessl and Y. Yang, Analysis of a Brownian particle moving in a time-dependent drift field, Asymptotic Analysis 27 (2001) 281–319.
A. Mandelbaum and W.A. Massey, Strong approximations for time dependent queues, Math. of Operations Research 20 (1995) 305–327.
W.A. Massey and W. Whitt, Networks of infinite server queues with nonstationary Poisson input, QUESTA 13 (1993) 183–250.
D. Mitra and A. Weiss, The transient behavior in: Erlang's model for large trunk groups and various traffic conditions, in Proceedings 12th International Teletraffic Congress (Torino, Italy, June 1988) pp. 5.1B4.1–5.1B4.8.
G.F. Newell, Queues with time-dependent arrival rates I: The transition through saturation, J. Applied Probability 5 (1968) 436–451.
G.F. Newell, Queues with time-dependent arrival rates II: The maximum queue and the return to equilibrium, J. Applied Probability 5 (1968) 579–590.
G.F. Newell, Queues with time-dependent arrival rates III: A mild rush hour, J. Applied Probability 5 (1968) 591–606.
I. Stakgold, Boundary Value Problems of Mathematical Physics, Volume I (MacMillan, New York, 1967).
S. Xie and C. Knessl, On the transient behavior of the Erlang loss model: Heavy usage asymptotics, SIAM J. Appl. Math. 53 (1993) 555–599.
Y. Yang and C. Knessl, Asymptotic analysis of the M/G/1 queue with a time-dependent arrival rate, QUESTA 26 (1997) 23–68.
Author information
Authors and Affiliations
Corresponding author
Additional information
AMS subject classification: 60K25,34E10
Supported in part by NSF grants DMS-99-71656 and DMS-02-02815
Rights and permissions
About this article
Cite this article
Knessl, C., Yang, Y.P. On the Erlang Loss Model with Time Dependent Input. Queueing Syst 52, 49–104 (2006). https://doi.org/10.1007/s11134-006-3753-1
Issue Date:
DOI: https://doi.org/10.1007/s11134-006-3753-1