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On the Erlang Loss Model with Time Dependent Input

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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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL, 60607-7045, USA

    Charles Knessl

  2. Department of Mathematics, University of St. Thomas, St. Paul, MN, 55105-1079, USA

    Yongzhi Peter Yang

Authors
  1. Charles Knessl
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  2. Yongzhi Peter Yang
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Correspondence to Charles Knessl.

Additional information

AMS subject classification: 60K25,34E10

Supported in part by NSF grants DMS-99-71656 and DMS-02-02815

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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

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  • DOI: https://doi.org/10.1007/s11134-006-3753-1

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