Page 187 - A First Course In Stochastic Models

180 CONTINUOUS-TIME MARKOV CHAINS

(c) What is the long-run average number of cars in the station?
(d) What is the long-run fraction of potential customers that are lost?
4.5 A production hall contains a fast machine and a slow machine to process incoming
orders. Orders arrive according to a Poisson process with rate λ. An arriving order that finds
both machines occupied is rejected. Unless both machines are occupied, an arriving order
is assigned to the fast machine if available; otherwise, the order is assigned to the slow
machine. The processing time of an order is exponentially distributed with mean 1/µ 1 at
the fast machine and mean 1/µ 2 at the slow machine. It is not possible to transfer an order
from the slow machine to the fast machine.
(a) Formulate a continuous-time Markov chain to analyse the situation in the production
hall. Specify the state variable(s) and the transition rate diagram
(b) Specify the equilibrium equations for the state probabilities. What is the long-run
fraction of time that the fast (slow) machine is used? What is the long-run fraction of
incoming orders that are lost?
4.6 In Gotham City there is a one-man taxi company. The taxi company has a stand at the
railway station. Potential customers arrive according to a Poisson process with an average
of four customers per hour. The taxi leaves the station immediately a customer arrives. A
potential customer finding no taxi present waits until the taxi arrives only if there are less
than three other customers waiting; otherwise, the customer goes elsewhere for alternative
transport. If the taxi returns to the stand and finds waiting customers, it picks up all waiting
customers and leaves. The amount of time needed to return to the stand has an exponential
distribution with mean 1/µ i when the taxi leaves the stand with i customers, i = 1, 2, 3.
(a) Formulate a continuous-time Markov chain to analyse the situation at the taxi stand.
Specify the state variable(s) and the transition rate diagram.
(b) What is the long-run fraction of time the taxi waits idle at the taxi stand? What is the
long-run fraction of potential customers who go elsewhere for transport?
4.7 A container terminal has a single unloader to unload trailers which bring loads of
containers. The unloader can serve only one trailer at a time and the unloading time has
an exponential distribution with mean 1/µ 1 . After a trailer has been unloaded, the trailer
leaves but the unloader needs an extra finishing time for the unloaded containers before
the unloader is available to unload another trailer. The finishing time has an exponential
distribution with mean 1/µ 2 . A leaving trailer returns with the next load of containers after
an exponentially distributed time with mean 1/λ. There are a finite number of N unloaders
active at the terminal.
(a) Formulate a continuous-time Markov chain to analyse the situation at the container
terminal. Specify the state variable(s) and the transition rate diagram.
(b) What is the long-run fraction of time the unloader is idle? What is the long-run
average number of trailers unloaded per time unit?
(c) What is the long-run average number of trailers waiting to be unloaded? What is the
long-run average waiting time per trailer?
(d) Write a computer program to compute the performance measures in (b) and (c) for
the numerical data N = 10, µ 1 = 1/3, µ 2 = 2 and λ = 1/50.
4.8 Messages for transmission arrive at a communication channel according to a Poisson
process with rate λ. The channel can transmit only one message at a time. The transmission
time is exponentially distributed with mean 1/µ. The following access control rule is used.
A newly arriving message is accepted as long as less than R other messages are present at
the communication channel (including any message in transmission). As soon as the number
of messages in the system has dropped to r, newly arriving messages are again admitted to
the transmission channel. The control parameters r and R are given integers with 0 ≤ r < R.
(a) Formulate a continuous-time Markov chain to analyse the situation at the communi-
cation channel. Specify the state variable(s) and the transition rate diagram.