We consider a variant of M/M/1 where customers arrive singly or in pairs. Each single and one member of each pair is called primary;the other member of each pair is called secondary. Each primary joins the queue upon ...We consider a variant of M/M/1 where customers arrive singly or in pairs. Each single and one member of each pair is called primary;the other member of each pair is called secondary. Each primary joins the queue upon arrival. Each secondary is delayed in a separate area, and joins the queue when “pushed” by the next arriving primary. Thus each secondary joins the queue followed immediately by the next primary. This arrival/delay mechanism appears to be new in queueing theory. Our goal is to obtain the steady-state probability density function (pdf) of the workload, and related quantities of interest. We utilize a typical sample path of the workload process as a physical guide, and simple level crossing theorems, to derive model equations for the steady-state pdf. A potential application is to the processing of electronic signals with error free components and components that require later confirmation before joining the queue. The confirmation is the arrival of the next signal.展开更多
This paper considers a Geo/Geo/1 queueing system with infinite capacity, in which the service rate changes depending on the workload. Initially, when the number of customers in the system is less than a certain thresh...This paper considers a Geo/Geo/1 queueing system with infinite capacity, in which the service rate changes depending on the workload. Initially, when the number of customers in the system is less than a certain threshold L, low service rate is provided for cost saving. On the other hand, the high service rate is activated as soon as L customers accumulate in the system and such service rate is preserved until the system becomes completely empty even if the number of customers falls below L. The steady-state probability distribution and the expected number of customers in the system are derived. Through the first-step argument, a recursive algorithm for computing the first moment of the conditional sojourn time is obtained. Furthermore, employing the results of regeneration cycle analysis, the direct search method is also implemented to determine the optimal value of L for minimizing the long-run average cost rate function.展开更多
文摘We consider a variant of M/M/1 where customers arrive singly or in pairs. Each single and one member of each pair is called primary;the other member of each pair is called secondary. Each primary joins the queue upon arrival. Each secondary is delayed in a separate area, and joins the queue when “pushed” by the next arriving primary. Thus each secondary joins the queue followed immediately by the next primary. This arrival/delay mechanism appears to be new in queueing theory. Our goal is to obtain the steady-state probability density function (pdf) of the workload, and related quantities of interest. We utilize a typical sample path of the workload process as a physical guide, and simple level crossing theorems, to derive model equations for the steady-state pdf. A potential application is to the processing of electronic signals with error free components and components that require later confirmation before joining the queue. The confirmation is the arrival of the next signal.
文摘This paper considers a Geo/Geo/1 queueing system with infinite capacity, in which the service rate changes depending on the workload. Initially, when the number of customers in the system is less than a certain threshold L, low service rate is provided for cost saving. On the other hand, the high service rate is activated as soon as L customers accumulate in the system and such service rate is preserved until the system becomes completely empty even if the number of customers falls below L. The steady-state probability distribution and the expected number of customers in the system are derived. Through the first-step argument, a recursive algorithm for computing the first moment of the conditional sojourn time is obtained. Furthermore, employing the results of regeneration cycle analysis, the direct search method is also implemented to determine the optimal value of L for minimizing the long-run average cost rate function.
