Some Inequalities in Queuing

Bounds are found for various measures of performance in certain classes of the GI//G1 queue. First, the mean wait in queue is found in terms of the mean and variance of the interarrival, service, and idle distributions. Bounds on the idle time moments lead to bounds on the mean wait and number in queue. The interarrival time distribution is then assumed to have mean residual life bounded above by 1/λ (λ = arrival rate); i.e., given a time t since the last arrival, the expected time to the next arrival is no more than 1/λ. With this assumption the mean number in queue (and hence system) is bounded to within (1 + p)/2 customers. Both upper and lower bounds are tight. The stronger assumption that, given time t since the last arrival, the probability an arrival occurs in the next ▵ t is nondecreasing in t, leads to bounds on the mean queue length to within (c²_a+p)/2, where c_a is the coefficient of variation of the arrival distribution. Again the bounds are tight. Specializing to the D/G/1 queue the mean queue length is found to within p/2 < 1/2 customer.