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On infinite server queues with batch arrivals

Published online by Cambridge University Press:  14 July 2016

D. N. Shanbhag
D. N. Shanbhag*
Affiliation:
Karnatak University
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Extract

The queueing system studied in this paper is the one in which

  1. (i) there are an infinite number of servers,

  2. (ii) initially (at t = 0) all the servers are idle,

  3. (iii) one server serves only one customer at a time and the service times are independent and identically distributed with distribution function B(t) (t > 0) and mean β(< ∞),

  4. (iv) the arrivals are in batches such that a batch arrives during (t, t + δt) with probability λ(t)δt + o(δt) (λ(t) > 0) and no arrival takes place during (t, t + δt) with the probability 1 –λ(t)δt + o(δt),

  5. (v) the batch sizes are independent and identically distributed with mean α(< ∞), and the probability that a batch size equals r is given by ar(r ≧ 1),

  6. (vi) the batch sizes, the service times and the arrivals are independent.

Type
Short Communications
Information
Journal of Applied Probability , Volume 3 , Issue 1 , June 1966 , pp. 274 - 279
Copyright
Copyright © Sheffield: Applied Probability Trust 

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References

[1] Downton, F. (1962) Congestion system with incomplete service. J. R. Statist. Soc. B 24, 107–111.Google Scholar
[2] Mirasol, N. M. (1963) The output of an M/G/8 queueing system is Poisson. Operat. Res. 11, 282–284.Google Scholar
[3] Shanbhag, D. N. (1964) On a problem of servicing a Poisson flow of demands. Ann. Math. Statist. 35, 461–462 (abstract).Google Scholar
[4] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press.Google Scholar