The Queue G/G/1 and Associated Random Walk

Abstract

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 − t_n−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

$$ {W_{n + 1}} = \max \left( {0,{X_{n + 1}} + {W_n}} \right),\;\;\;{I_{n + 1}} = - \min \left( {0,{X_{n + 1}} + {W_n}} \right). $$

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