Show that for a G/G/1 queue, the time-average wait in the system is the same as limn→∞ E[Wn]. Hint: Consider an integer renewal counting process {M(n); n 0} where M(n) is the number of renewals in the G/G/1 process of Section 5.5.3 that have occurred by the nth arrival. Show that this renewal process has a span of 1. Then consider {Wn; n 1} as […]

This problem is designed to give you an alternate way of looking at ensembleaverages for renewal-reward problems. First we find an exact expression for Pr SN(t) > s . We find this for arbitrary s and t, 0 This problem is designed to give you an alternate way of looking at ensembleaverages for… Get Your Custom Essay on this Assignment/Homework Order Essay  

This problem provides another way of treating ensemble-averages for renewal reward problems. Assume for notational simplicity that X is a continuous random variable. This problem provides another way of treating ensemble-averages for renewal reward problems…. Get Your Custom Essay on this Assignment/Homework Order Essay

Let {N1(t); t>0} and {N2(t); t>0} be independent renewal counting processes. Assume that each has the same distribution function F(x) for interarrival intervals and assume that a density f(x) exists for the interarrival intervals. Let {N 1 (t); t>0} and {N 2 (t); t>0} be independent renewal counting processes. Assume that… Get Your Custom Essay on this Assignment/Homework Order Essay  

A large system is controlled by n identical computers. Each computer independently alternates between an operational state and a repair state. The duration of the operational state, from completion of one repair until the next need for repair, is a random variable X with finite expected duration E [X]. The time required to repair a computer is an exponentially distributed random variable with density λe-λt. […]

Consider a sequence X1, X2, . . . of IID binary random variables. Let p and 1 p denote Pr{Xm = 1} and Pr{Xm = 0} respectively. A renewal is said to occur at time m if Xm-1 = 0 and Xm = 1. Consider a sequence X 1 , X 2 , . . . of IID binary random variables. Let p and 1 […]

An M/G/1 queue has arrivals at rate and a service time distribution given by FY(y). Assume that λ Z(z) be the distribution of the inter-renewal intervals and let E [Z] be the mean inter-renewal interval. An M/G/1 queue has arrivals at rate and a service time distribution given by F Y (y). Assume that… Get Your Custom Essay on this Assignment/Homework Order Essay a) Find […]

Consider the same setup as in Exercise 5.28 except that now customers arrive according to a non-arithmetic renewal process independent of the bus arrival process. Let 1/λ be the expected inter-renewal interval for the customer renewal process. Assume that both renewal processes are in steady state (i.e., either we look only at t >> 0, or we assume that they are equilibrium processes). Given that […]

Let {N1(t); t > 0} be a Poisson counting process of rate λ. Assume that the arrivals from this process are switched on and off by arrivals from a non-arithmetic renewal counting process {N2(t); t > 0} (see figure below). The two processes are independent Let {N 1 (t); t > 0} be a Poisson counting process of rate ?. Assume that the arrivals from… […]

a) Find limt→∞{E [N(t)]t/} for a renewal counting process {N(t); t > 0} with inter-renewal times {Xi;i ≥ 1}. a) Find lim t?8 {E [N(t)]t/ } for a renewal counting process {N(t); t > 0} with inter-renewal… Get Your Custom Essay on this Assignment/Homework Order Essay b) Evaluate your result for the case in which X is an exponential random variable (you already know what […]