91Ó°ÊÓ

A doctor has scheduled two appointments, one at \(1 \mathrm{P} . \mathrm{M}\). and the other at \(1: 30 \mathrm{P.M}\). The amounts of time that appointments last are independent exponential random variables with mean 30 minutes. Assuming that both patients are on time, find the expected amount of time that the \(1: 30\) appointment spends at the doctor's office.

A certain scientific theory supposes that mistakes in cell division occur according to a Poisson process with rate \(2.5\) per year, and that an individual dies when 196 such mistakes have occurred. Assuming this theory, find (a) the mean lifetime of an individual, (b) the variance of the lifetime of an individual. Also approximate (c) the probability that an individual dies before age \(67.2\), (d) the probability that an individual reaches age 90 ,

Let \(\\{N(t), t \geqslant 0\\}\) be a Poisson process with rate \(\lambda\). Let \(S_{n}\) denote the time of the \(n\) th event. Find (a) \(E\left[S_{4}\right]\), (b) \(E\left[S_{4} \mid N(1)=2\right]\) (c) \(E[N(4)-N(2) \mid N(1)=3]\)

Customers arrive at a two-server service station according to a Poisson process with rate \(\lambda .\) Whenever a new customer arrives, any customer that is in the system immediately departs. A new arrival enters service first with server 1 and then with server 2\. If the service times at the servers are independent exponentials with respective rates \(\mu_{1}\) and \(\mu_{2}\), what proportion of entering customers completes their service with server 2?

Consider a two-server parallel queuing system where customers arrive according to a Poisson process with rate \(\lambda\), and where the service times are exponential with rate \mu. Moreover, suppose that arrivals finding both servers busy immediately depart without receiving any service (such a customer is said to be lost), whereas those finding at least one free server immediately enter service and then depart when their service is completed. (a) If both servers are presently busy, find the expected time until the next customer enters the system. (b) Starting empty, find the expected time until both servers are busy. (c) Find the expected time between two successive lost customers.

Events occur according to a Poisson process with rate \(\lambda .\) Each time an event occurs, we must decide whether or not to stop, with our objective being to stop at the last event to occur prior to some specified time \(T\), where \(T>1 / \lambda\). That is, if an event occurs at time \(t, 0 \leqslant t \leqslant T\), and we decide to stop, then we win if there are no additional events by time \(T\), and we lose otherwise. If we do not stop when an event occurs and no additional events occur by time \(T\), then we lose. Also, if no events occur by time \(T\), then we lose. Consider the strategy that stops at the first event to occur after some fixed time \(s, 0 \leqslant s \leqslant T\). (a) Using this strategy, what is the probability of winning? (b) What value of \(s\) maximizes the probability of winning? (c) Show that one's probability of winning when using the preceding strategy with the value of \(s\) specified in part (b) is \(1 / e\).

If an individual has never had a previous automobile accident, then the probability he or she has an accident in the next \(h\) time units is \(\beta h+o(h) ;\) on the other hand, if he or she has ever had a previous accident, then the probability is \(\alpha h+o(h) .\) Find the expected number of accidents an individual has by time \(t\).

Teams 1 and 2 are playing a match. The teams score points according to independent Poisson processes with respective rates \(\lambda_{1}\) and \(\lambda_{2} .\) If the match ends when one of the teams has scored \(k\) more points than the other, find the probability that team 1 wins. Hint: Relate this to the gambler's ruin problem.

Consider a single server queuing system where customers arrive according to a Poisson process with rate \(\lambda\), service times are exponential with rate \(\mu\), and customers are served in the order of their arrival. Suppose that a customer arrives and finds \(n-1\) others in the system. Let \(X\) denote the number in the system at the moment that customer departs. Find the probability mass function of \(X\). Hint: Relate this to a negative binomial random variable.

Events occur according to a Poisson process with rate \(\lambda=2\) per hour. (a) What is the probability that no event occurs between \(8 \mathrm{P} . \mathrm{M} .\) and \(9 \mathrm{P.M.?}\) (b) Starting at noon, what is the expected time at which the fourth event occurs? (c) What is the probability that two or more events occur between \(6 \mathrm{P.M}\). and 8 P.M.?