The Queue G/G/1 and Associated Random Walk

  • N. U. Prabhu
Part of the Applications of Mathematics book series (SMAP, volume 15)


We consider the single-server queueing system where successive customers arrive at the epochs t0(= 0), t1, t2, ..., and demand services times υ1, υ2, ... . The inter-arrival times are then given by u n = t n tn−1 (n ≥ 1). Let X k = υ k u k (k ≥ 1), and S0 ≡ 0, S n = X1 + X2 + ⋯ + X n (n ≥ 1). We assume that the X k are mutually independent random variables with a common distribution; the basic process underlying this queueing model is the random walk {S n }. To see this, let W n be the waiting time of the nth customer and I n the idle period (if any) that just terminates upon the arrival of this customer. Then clearly for n ≥ 0
$$ {W_{n + 1}} = \max \left( {0,{X_{n + 1}} + {W_n}} \right),\;\;\;{I_{n + 1}} = - \min \left( {0,{X_{n + 1}} + {W_n}} \right). $$


Random Walk Queue Length Busy Period Idle Period Queue Length Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag New York Inc. 1980

Authors and Affiliations

  • N. U. Prabhu
    • 1
  1. 1.School of Operations ResearchCornell UniversityIthacaUSA

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