91Ó°ÊÓ

A population of organisms consists of both male and female members. In a small colony any particular male is likely to mate with any particular female in any time interval of length \(h\), with probability \(\lambda h+o(h) .\) Each mating immediately produces one offspring, equally likely to be male or female. Let \(N_{1}(t)\) and \(N_{2}(t)\) denote the number of males and females in the population at \(t .\) Derive the parameters of the continuous-time Markov chain \(\left\\{N_{1}(t), N_{2}(t)\right\\}\), i.e., the \(v_{i}, P_{i j}\) of Section \(6.2\).

Suppose that a one-celled organism can be in one of two states-either \(A\) or \(B\). An individual in state \(A\) will change to state \(B\) at an exponential rate \(\alpha ;\) an individual in state \(B\) divides into two new individuals of type \(A\) at an exponential rate \(\beta .\) Define an appropriate continuous- time Markov chain for a population of such organisms and determine the appropriate parameters for this model.

Consider two machines that are maintained by a single repairman. Machine \(i\) functions for an exponential time with rate \(\mu_{i}\) before breaking down, \(i=1,2 .\) The repair times (for either machine) are exponential with rate \(\mu .\) Can we analyze this as a birth and death process? If so, what are the parameters? If not, how can we analyze it?

Potential customers arrive at a single-server station in accordance with a Poisson process with rate \(\lambda .\) However, if the arrival finds \(n\) customers already in the station, then he will enter the system with probability \(\alpha_{n}\). Assuming an exponential service rate \(\mu\), set this up as a birth and death process and determine the birth and death rates.

Consider a birth and death process with birth rates \(\lambda_{i}=(i+1) \lambda, i \geqslant 0\), and death rates \(\mu_{i}=i \mu, i \geqslant 0\) (a) Determine the expected time to go from state 0 to state 4 . (b) Determine the expected time to go from state 2 to state 5 . (c) Determine the variances in parts (a) and (b).

Consider two machines. Machine \(i\) operates for an exponential time with rate \(\lambda_{i}\) and then fails; its repair time is exponential with rate \(\mu_{i}, i=1,2 .\) The machines act independently of each other. Define a four-state continuous-time Markov chain that jointly describes the condition of the two machines. Use the assumed independence to compute the transition probabilities for this chain and then verify that these transition probabilities satisfy the forward and backward equations.

Each individual in a biological population is assumed to give birth at an exponential rate \(\lambda\), and to die at an exponential rate \(\mu .\) In addition, there is an exponential rate of increase \(\theta\) due to immigration. However, immigration is not allowed when the population size is \(N\) or larger. (a) Set this up as a birth and death model. (b) If \(N=3,1=\theta=\lambda, \mu=2\), determine the proportion of time that immigration is restricted.

A small barbershop, operated by a single barber, has room for at most two customers. Potential customers arrive at a Poisson rate of three per hour, and the successive service times are independent exponential random variables with mean \(\frac{1}{4}\) hour. (a) What is the average number of customers in the shop? (b) What is the proportion of potential customers that enter the shop? (c) If the barber could work twice as fast, how much more business would he do?

Potential customers arrive at a full-service, one-pump gas station at a Poisson rate of 20 cars per hour. However, customers will only enter the station for gas if there are no more than two cars (including the one currently being attended to) at the pump. Suppose the amount of time required to service a car is exponentially distributed with a mean of five minutes. (a) What fraction of the attendant's time will be spent servicing cars? (b) What fraction of potential customers are lost?

A service center consists of two servers, each working at an exponential rate of two services per hour. If customers arrive at a Poisson rate of three per hour, then, assuming a system capacity of at most three customers, (a) what fraction of potential customers enter the system? (b) what would the value of part (a) be if there was only a single server, and his rate was twice as fast (that is, \(\mu=4)\) ?