The Queue GI/G/1

We consider the single-server queueing system where successive customers arrive at the epochs t₀ (= 0), t₁, t₂,... and demand services times v₁, v₂,.... The interarrival times are then given by u _n = t _n — t _n-1 (n≥1). Let X_k = V_k — u_k (k ≥ 1), and So = 0, S _n = X ₁ +X ₂ + … + X _n (n ≥ 1). We assume that the X_kare 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≥ 01$$ W_{n + 1} = \left( {X_{n + 1} + W_n } \right)^ + , I_{n + 1} = \left( {X_{n + 1} + W_n } \right)^ - $$.