Find the probability density of ui

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Consider a variation of an M/G/1 queueing system in which there is no facility to save waiting customers. Assume customers arrive according to a Poisson process of rate λ. If the server is busy, the customer departs and is lost forever; if the server is not busy, the customer enters service with a service time CDF denoted by F(y). Successive service times (for those customers that are served) are IID and independent of arrival times. Assume that customer number 0 arrives and enters service at time = 0.

(a) Show that the sequence of times S1, S2, ... at which successive successful customers enter service are the renewal times of a renewal process. Show that each inter-renewal interval XSSi-1  (where S0  = 0) is the sum of two  independent rv s, YUwhere Yis the ith service time; find the probability density of Ui.

(b) Assume that a reward (actually a cost in this case) of one unit is incurred for each customer turned away. Sketch the expected reward function as a function of time for the sample function of inter-renewal intervals and service intervals shown below; the expectation is to be taken over those (unshown) arrivals of customers that must be turned away.

712_Inter-Renewal Intervals and Service Intervals.png

(c) Let ( R(τ )dτ denote the accumulated reward (i.e., cost) from 0 to and find the limit as → ∞ of (1/t) ( R(τ )dτ . Explain (without any attempt to be rigorous or formal) why this limit exists with probability 1.

(d) In the limit of large t, find the expected reward from time until the next renewal. Hint: Sketch this expected reward as a function of for a given sample of inter-renewal intervals and service intervals; then find the time average.

(e) Now assume that the arrivals are deterministic, with the first arrival at time 0 and the nth arrival at time - 1. Does the sequence of times S1, S2, ... at which subsequent customers start service still constitute the renewal times of a renewal process? Draw a sketch of arrivals, departures, and service time intervals. Again find t limt→∞((R(τ dτ )/t.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.

Reference no: EM13909444

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