Consider a barbershop with two barbers and two waiting chairs. Customers arrive at a rate
of 5 per hour. Customers arriving to a fully occupied shop leave without being served. Find
the stationary distribution for the number of customers in the shop, assuming that the service
rate for each barber is 2 customers per hour.

t. Consider a barbershop with one barber who can cut hair at rate 4 and three waiting chairs.
Customers arrive at a rate of 5 per hour.

(a) Argue that this new set-up will result in fewer lost customers than the scheme from
Problem 1.
(b) Compute the increase in the number of customers served per hour.

L. A taxi company has three cabs. Calls come in to the dispatcher at times of a Poisson process
with rate 2 per hour. Suppose that each requires an exponential amount of time with mean
20 minutes, and that callers will hang up if they hear there are no cabs available.

(a) What is the probability all three cabs are busy when a call comes in?
(b) In the long run, on the average how many customers are served per hour?

Brad’s relationship with his girl friend Angelina changes between Amorous, Bickering, Confu-
sion, and Depression according to the following transition rates when t is the time in months.

A B C D
A -4 3 1 0
Q _ B 4 -6 2 0
C 2 3 -6 1
D 0 0 2 -2
(a) Find the long run fraction of time he spends in these four states.
(b) Does the chain satisfy the detailed balance condition?
(c) They are amorous now. What is the expected amount of time until depression sets in?

A submarine has three navigational devices but can remain at sea if at least two are working.
Suppose that the failure times are exponential with means 1 year, 1.5 years, and 3 years.
Formulate a Markov chain with states 0 = all parts working, 1 = part 1 failed, 2 = part 2
failed, 3 = part 3 failed, 4 = two parts failed. Compute EON/21] (the expected hitting time of
state 4 starting from state 0) to determine the average length of time the boat can remain at
sea.

5. Consider a production system consisting of a machine center followed by an inspection station.
Arrivals from outside the system occur only at the machine center and follow a Poisson process
with rate /. The machine center and inspection station are each single-server operations with
rates #1 and p2. Suppose that each item independently passes inspection with probability
p. When an object fails inspection it is sent to the machine center for reworking. Find
the conditions on the parameters that are necessary for the system to have a stationary
distribution.

7. At registration at a very small college, students arrive at the English table at rate 10 and
at the Math table at rate 5. A student who completes service at the English table goes to
the Math table with probability 1/4 and to the cashier with probability 3/4. A student who
completes service at the Math table goes to the English table with probability 2/5 and to the
cashier with probability 3/5. Students who reach the cashier leave the system after they pay.
Suppose that the service rates for the English table, Math table, and cashier are 25, 30, and
20, respectively. All three stations, the English table, the Math table, and the cashier have
a single server. Find the stationary distribution and the average time students spend in the
system.

Consider a barbershop with two barbers and two waiting chairs. Customers arrive at a rate
of 5 per hour. Customers arriving to a fully occupied shop leave without being served. Find
the stationary distribution for the number of customers in the shop, assuming that the service
rate for each barber is 2 customers per hour.

t. Consider a barbershop with one barber who can cut hair at rate 4 and three waiting chairs.
Customers arrive at a rate of 5 per hour.

(a) Argue that this new set-up will result in fewer lost customers than the scheme from
Problem 1.
(b) Compute the increase in the number of customers served per hour.

L. A taxi company has three cabs. Calls come in to the dispatcher at times of a Poisson process
with rate 2 per hour. Suppose that each requires an exponential amount of time with mean
20 minutes, and that callers will hang up if they hear there are no cabs available.

(a) What is the probability all three cabs are busy when a call comes in?
(b) In the long run, on the average how many customers are served per hour?

Brad’s relationship with his girl friend Angelina changes between Amorous, Bickering, Confu-
sion, and Depression according to the following transition rates when t is the time in months.

A B C D
A -4 3 1 0
Q _ B 4 -6 2 0
C 2 3 -6 1
D 0 0 2 -2
(a) Find the long run fraction of time he spends in these four states.
(b) Does the chain satisfy the detailed balance condition?
(c) They are amorous now. What is the expected amount of time until depression sets in?

A submarine has three navigational devices but can remain at sea if at least two are working.
Suppose that the failure times are exponential with means 1 year, 1.5 years, and 3 years.
Formulate a Markov chain with states 0 = all parts working, 1 = part 1 failed, 2 = part 2
failed, 3 = part 3 failed, 4 = two parts failed. Compute EON/21] (the expected hitting time of
state 4 starting from state 0) to determine the average length of time the boat can remain at
sea.

5. Consider a production system consisting of a machine center followed by an inspection station.
Arrivals from outside the system occur only at the machine center and follow a Poisson process
with rate /. The machine center and inspection station are each single-server operations with
rates #1 and p2. Suppose that each item independently passes inspection with probability
p. When an object fails inspection it is sent to the machine center for reworking. Find
the conditions on the parameters that are necessary for the system to have a stationary
distribution.

7. At registration at a very small college, students arrive at the English table at rate 10 and
at the Math table at rate 5. A student who completes service at the English table goes to
the Math table with probability 1/4 and to the cashier with probability 3/4. A student who
completes service at the Math table goes to the English table with probability 2/5 and to the
cashier with probability 3/5. Students who reach the cashier leave the system after they pay.
Suppose that the service rates for the English table, Math table, and cashier are 25, 30, and
20, respectively. All three stations, the English table, the Math table, and the cashier have
a single server. Find the stationary distribution and the average time students spend in the
system.