Page 236 - A First Course In Stochastic Models

EXERCISES 229

5.21 Consider the following modification of Example 5.4.2. Instead of infinite-source input,
there is finite-source input for each of the two message types. The source of messages of type
j has M j users, where each user generates a new message after an exponentially distributed
think time with mean 1/λ j provided the user has no message in service at the communication
system. Assume the numerical data c = 10, M 1 = M 2 = 10, λ 1 = 3, λ 2 = 1, µ 1 = 4,
µ 2 = 1. Use continuous-time Markov chain analysis to compute the L-policy for which
the average throughput is maximal. Does the result change when the transmission times are
constant rather than exponentially distributed?
5.22 Suppose a production facility has M operating machines and a buffer of B standby
machines. Machines in operation are subject to breakdowns. The running times of the oper-
ating machines are independent of each other and have a common exponential distribution
with mean 1/λ. An operating machine that breaks down is replaced by a standby machine if
one is available. A failed machine immediately enters repair. There are ample repair facil-
ities so that any number of machines can be repaired simultaneously. The repair time of a
failed machine is assumed to have an exponential distribution with mean 1/µ. For given
values of µ, λ and M, demonstrate how to calculate the minimum buffer size B in order
to achieve that the long-run fraction of time that less than M machines are operating is no
more than a specific value β. Do you expect the answer to depend on the specific form of
the repair-time distribution?
5.23 Suppose a communication system has c transmission channels at which messages arrive
according to a Poisson process with rate λ. Each message that finds all of the c channels busy
is lost upon arrival, otherwise the message is randomly assigned to one of the free channels.
The transmission length of an accepted message has an exponential distribution with mean
1/µ. However, each separate channel is subject to a randomly changing environment that
influences the transmission rate of the channel. Independently of each other, the channels
alternate between periods of good condition and periods of bad condition. These alternating
periods are independent of each other and have exponential distributions with means 1/γ g
and 1/γ b . The transmission rate of a channel being in good (bad) condition is σ g (σ b ). Set
up the balance equations for calculating the fraction of messages that are lost. Noting that
σ = (σ b γ g + σ g γ b )/(γ g + γ b ) is the average transmission rate used by a channel, make
some numerical comparisons with the case of a fixed transmission rate σ.
5.24 Jobs have to undergo tooling at two stations, 1 and 2, which are linked in series. New
jobs arrive at station 1 according to a Poisson process with rate λ. At station 1 they undergo
their first tooling. Upon completion of the tooling at station 2, there is a given probability
p that both toolings have to be done anew. In this case the job rejoins the queue at station
1, otherwise the job leaves the system. The handling times of a job at stations 1 and 2 are
independent random variables having exponential distributions with respective means 1/µ 1
and 1/µ 2 . Each station can handle only one job at a time. What is the long-run average
amount of time spent in the system by a newly arriving job?
5.25 Consider a closed queueing network as in Section 5.6.2. Assume now that the service
rate at station i is a function µ i (n i ) of the number (n i ) of customers present at station i.
Verify that the product-form solution is given by

 
K n i

n i
p(n 1 , . . . n K ) = C  λ / µ i (l)  .
i
i=1 l=1
5.26 Consider the M/G/1 queue with Erlangian services from Example 5.5.1. Define the
∞ ∞
f
β
j j
generating functions β(z) = j=1 j z and F(z) = j=0 j z . Let R be the convergence
j
radius of the series j=1 j z . It is assumed that R > 1.
∞
β