Page 234 - A First Course In Stochastic Models

EXERCISES 227

exactly one unit is placed each time the on-hand inventory decreases by one unit. The lead
times of the replenishments are independent and identically distributed random variables
with mean τ. Establish an equivalence with the Erlang loss model and give expressions for
the long-run average on-hand inventory and the long-run fraction of demand that is lost.
5.14 In an electronic system there are c elements of a crucial component connected in
parallel to increase the reliability of the system. Each component is switched on and the
lifetimes of the components have an exponential distribution with mean 1/α. The lifetimes
of the components are independent of each other. The electronic system is working as long
as at least one of the components is functioning, otherwise the system is down. A component
that fails is replaced by a new one. It takes an exponentially distributed time with mean 1/β
to replace a failed component. Only one failed component can be replaced at a time.
(a) Use a continuous-time Markov chain to calculate the long-run fraction of time the
system is down. Specify the transition rate diagram first.
(b) Does the answer in (a) change when the replacement time of a failed component has a
general probability distribution with mean 1/α? (Hint: compare the transition rate diagram
with the transition rate diagram in the Erlang loss model.)
5.15 Reconsider Exercise 5.14 but this time assume there are ample repairmen to replace
failed components.
(a) Use a continuous-time Markov chain to calculate the long-run fraction of time the
system is down. Specify the transition rate diagram first.
(b) What happens to the answer in (a) when the replacement time is fixed rather than
exponentially distributed? (Hint: compare the transition rate diagram with the transition rate
diagram in the Engset loss model.)
5.16 Suppose you have two groups of servers each without waiting room. The first group
consists of c 1 identical servers each having an exponential service rate µ 1 and the second
group consists of c 2 identical servers each having an exponential service rate µ 2 . Customers
for group i arrive according to a Poisson process with rate λ i (i = 1, 2). A customer who
finds all servers in his group busy upon arrival is served by a server in the other group,
provided one is free, otherwise the customer is lost. Show how to calculate the long-run
fraction of customers lost.
5.17 Consider a conveyor system at which items for processing arrive according to a Poisson
process with rate λ. The service requirements of the items are independent random variables
having a common exponential distribution with mean 1/µ. The conveyor system has two
work stations 1 and 2 that are placed according to this order along the conveyor. Workstation
i consists of s i identical service channels, each having a constant processing rate of σ i
(i = 1, 2); that is, an item processed at workstation i has an average processing time of
1/(σ i µ). Both workstations have no storage capacity and each service channel can handle
only one item at a time. An arriving item is processed by the first workstation in which a
service channel is free and is lost when no service channel is available at either of the stations.
Show how to calculate the fraction of items lost and solve for the numerical data λ = 10,
µ = 1, σ 1 = 2, σ 2 = 1.5, s 1 = 5 and s 2 = 5 (Answer: 0.0306). Verify experimentally that
the loss probability is nearly insensitive to the distributional form of the service requirement
(e.g. compute the loss probability 0.0316 for the data when the service requirement has an
H 2 distribution with balanced means and a squared coefficient of variation of 4).
5.18 Consider a stochastic service system with Poisson arrivals at rate λ and two different
groups of servers, where each arriving customer simultaneously requires a server from both
groups. An arrival not finding that both groups have a free server is lost and has no further
influence on the system. The ith group consists of s i identical servers (i = 1, 2) and each
server can handle only one customer at a time. An entering customer occupies the two
assigned servers from the groups 1 and 2 during independently exponentially distributed
times with respective means 1/µ 1 and 1/µ 2 . Show how to calculate the loss probability