Performance and Constraint Optimization of M/G/1 Queue with Randomized Startup Time and the Bi-Level(p,N1,N2)-Policy
This paper considers an M/G/1 queue with randomized startup time under the control of the bi-level(p,N1,N2)-policy,where the bi-level(p,N1,N2)-policy means that if the number of customs in the system is equal to a given low integer threshold value N1(≥ 1),the server starts the system with probability p(0 ≤ p ≤ 1)or still left with probability 1-p.When the number of customs in the system reaches a higher integer threshold value N2(N2 ≥ N1),the server starts the system immediately.Meanwhile,it takes a random length of time to start the system.When the system startup is completed,the server begins service immediately.Using the renewal process theory,the total probability decomposition technique and the Laplace transformation,we discuss the transient and steady-state distributions of the queue size.Both the expressions of the Laplace transform of the transient queue-length distribution with respect to time t and the recursive expressions of the steady-state queue length distribution are obtained.Meanwhile,some other important queueing performance indicators of the system are derived.Furthermore,we illustrate the important application of the recursive expressions of the steady-state queue length distribution in the system capacity design by a numerical example.Finally,employing the renewal reward theorem,the recursive expression of the long-run expected cost per unit time of the system is obtained.Moreover,under assuming that the service time and set-up time obey PH distributions and the limit of the average waiting time of customer,numerical examples are presented to discuss the bi-level optimal control strategy(N1*,N2*)for minimizing the long-run expected cost per unit time of the system as well as the influence of parameter p on the long-run expected cost per unit time of the system and expected waiting time of customer.
M/G/1 queuerandomized startup timebi-level(p,N1,N2)-policyqueue length distributionoptimal control strategy
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