Abstract
Consider aG/M/s/r queue, where the sequence{A n } ∞n=−∞ of nonnegative interarrival times is stationary and ergodic, and the service timesS n are i.i.d. exponentially distributed. (SinceA n =0 is possible for somen, batch arrivals are included.) In caser < ∞, a uniquely determined stationary process of the number of customers in the system is constructed. This extends corresponding results by Loynes [12] and Brandt [4] forr=∞ (withρ=ES0/EA0<s) and Franken et al. [9], Borovkov [2] forr=0 ors=∞. Furthermore, we give a proof of the relation min(i, s)¯p(i)=ρp(i−1), 1⩽i⩽r + s, between the time- and arrival-stationary probabilities¯p(i) andp(i), respectively. This extends earlier results of Franken [7], Franken et al. [9].
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Brandt, A. On stationary queue length distributions for G/M/s/r queues. Queueing Syst 2, 321–332 (1987). https://doi.org/10.1007/BF01150044
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DOI: https://doi.org/10.1007/BF01150044