What is the expected time until the DSCTMC leaves state 1?
IE 425 Homework 9
Submit on Tuesday, 12/3
1. Report your notebook score for Midterm Exam 2 along with a picture as the proof.
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Order Paper Now2. (11 pts) Consider a Discrete State Continuous Time Markov Chain (DSCTMC) defined on
Ω = {1, 2, 3} with generator matrix G:
G =
−6 2 41 −2 1
3 1 −4
Suppose the DSCTMC is in state 1.
(a) What is the expected time until the DSCTMC leaves state 1?
(b) What is the probability that the DSCTMC will jump to state 2 after it leaves state 1?
In Problem 3 ∼ Problem 10, model the systems as DSCTMCs. For each DSCTMC: (a) Define the states of the DSCTMC and write down their holding time distributions.
(b) Write down the transition probability matrix P of the jump chain of the DSCTMC.
(c) Write down the generator matrix G of the DSCTMC.
(d) Draw the transition rate diagram of the DSCTMC.
3. (11 pts) A machine, once in production mode, operates continuously until an alarm signal is
generated. The time up to the alarm signal is an exponential random variable with λ1 = 1. Sub-
sequent to the alarm signal, the machine is tested for an exponentially distributed amount of time
with λ2 = 5. The test results are positive, with probability 0.5, in which case the machine returns
to production mode, or negative, with probability 0.5, in which case the machine is taken for repair.
The duration of the repair is exponentially distributed with λ3 = 3. We assume that the above
mentioned random variables are all independent and also independent of the test results. Does the
long-run convergence theorem apply to this DSCTMC? Why? If so, what are the portions of time
that the DSCTMC spends in production mode, test mode, and repair mode, respectively?
4. (11 pts) Consider two machines that are maintained by a single repairman. Machine i functions
for an exponential time with rate µi before breaking down, i = 1, 2. The repair times (for either
machine) are exponential with rate λ.
5. (11 pts) M/M/1/∞ Queuing System Consider a food truck that sells lunch on the outskirts of a college campus. Customers arrive to the food truck according to a Poisson process with rate
λ (customers arrive one at a time). Customers are served by one cashier, service times follow an
exponential distribution with mean 1/µ. Assuming ρ = λ µ < 1, show that the stationary distribution
π of the M/M/1/∞ queuing system admits the following form:
πn =
( λ
µ
)n ( 1 −
λ
µ
) for n = 0, 1, 2, · · ·
6. (11 pts) M/M/c/∞ Queuing System A bank has c tellers. When a customer arrives, he goes to an empty teller (if there is one) or joins a single queue. Customers arrive following a Poisson process
with rate λ. Transaction times between a teller and a customer follow an exponential distribution
1
with mean 1/µ. Assuming ρ = λ cµ < 1, show that the stationary distribution π of the M/M/c/∞
queuing system admits the following form:
π0 =
[ 1 +
c−1∑ i=1
1
i!
( λ
µ
)i +
1
c!
( λ
µ
)c ( 1 −
λ
cµ
)−1]−1
πn =
1
n!
( λ
µ
)n π0 for 1 ≤ n ≤ c− 1
1
c!
( λ
µ
)c ( λ
cµ
)n−c π0 for n ≥ c
7. (11 pts) M/M/∞/∞ Queuing System Consider a self-service system where an unlimited number of servers are always available. Customers arrive following a Poisson process with rate λ. All
customers in the system at any instant are simultaneously being served, with each customer’s service
time following an exponential distribution with mean 1/µ. Show that the stationary distribution π
of the M/M/∞/∞ queuing system admits the following form:
πn = 1
n!
( λ
µ
)n e − λ
µ for n ≥ 0
Hint: Taylor expansion of exponential functions.
8. (11 pts) M/M/c/c Queuing System A telephone company owns a limited number c of transatlantic
telephone lines. When a customer wants to call overseas, he is assigned a line immediately provided
the lines are not all busy. If all lines are busy, customer is denied service and asked to try again later.
Calls arrive according to a Poisson process with rate λ. Each call has an exponentially distributed
length with mean 1/µ. Show that the stationary distribution π of the M/M/c/c queuing system
admits the following form:
π0 =
[ 1 +
c∑ i=1
1
i!
( λ
µ
)i]−1 πn =
1
n!
( λ
µ
)n π0 for 1 ≤ n ≤ c
9. (11 pts) M/M/c/k Queuing System Consider a manufacturing shop. Parts arrive according to
a Poisson process with rate λ. Shop contains c machines, allowing up to c parts to be processed
simultaneously. Shop has queue space for up to (k−c) other parts waiting in line when all machines are busy. Time required to process a part follows an exponential distribution with mean 1/µ. Show
that the stationary distribution π of the M/M/c/k queuing system admits the following form:
π0 =
[ 1 +
c−1∑ i=1
1
i!
( λ
µ
)i +
1
c!
( λ
µ
)c k∑ i=c
( λ
cµ
)i−c]−1
πn =
1
n!
( λ
µ
)n π0 for 1 ≤ n ≤ c− 1
1
c!
( λ
µ
)c ( λ
cµ
)n−c π0 for c ≤ n ≤ k
2
10. (12 pts) M/M/c/∞ Queuing System with a Finite Calling Population Suppose c maintenance personnel is responsible for keeping a set of N machines in operational order. Each maintainer
can repair a machine individually with an exponentially distributed amount of time with mean
1/µ. For each machine, the elapsed time between when it is returned to a serviceable condition
and when it next breaks down follows an exponential distribution with mean 1/λ. (Each machine
is considered a customer in the queueing system when it is down waiting to be repaired, when a
machine is operational it is outside the queuing system.) Show that the stationary distribution π
of the M/M/c/∞ queuing system with a finite Calling Population N(N > c) admits the following form:
π0 =
[ 1 +
c−1∑ i=1
N!
(N − i)!i!
( λ
µ
)i +
N∑ i=c
N!
(N − i)!c!ci−c
( λ
µ
)i]−1
πn =
N!
(N −n)!n!
( λ
µ
)n π0 for 0 ≤ n ≤ c− 1
N!
(N −n)!c!cn−c
( λ
µ
)n π0 for c ≤ n ≤ N
3
