Abstract
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.
Highlights
Each secondary joins the queue followed immediately by the primary. This arrival/delay mechanism appears to be new in queueing theory
The M/M/1 arrival/delay mechanism considered in this paper was introduced by Hlynka [1], who derived the Laplace transform of the busy period of the server, using the probabilistic interpretation of the Laplace transform
The busy period in that analysis included the idle times of the server while a secondary is being delayed
Summary
The M/M/1 arrival/delay mechanism considered in this paper was introduced by Hlynka [1], who derived the Laplace transform of the busy period of the server, using the probabilistic interpretation of the Laplace transform. An advantage of the level crossing method used here is that it focuses on the workload process in a concrete manner That is, it uses physical properties of a typical sample path of the workload process as a guide, and simple level crossing theorems, to formulate the model equations for the key probability distributions of the model. We select an arbitrary continuous subset in each sheet, having one boundary as a fixed level x in the state space of W t , e.g., x, , x 0 We use this concrete physical picture as a guide to balance the samplepath exit and entrance rates of the selected state-space subsets. All jumps due to an arrival on page 1 are distributed as E2 because both the delayed secondary and the arriving primary join the queue simultaneously
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