Page 189 - A First Course In Stochastic Models

182 CONTINUOUS-TIME MARKOV CHAINS

(a) Derive a recursion scheme for computing the limiting distribution of the number of
messages present and give an expression for the long-run average cost per time unit.
(b) Write a computer program for calculating the value of R which minimizes the average
cost and solve for the numerical data λ = 0.8, µ = 1, σ 1 = 1, σ 2 = 1.5, h = 1, r 0 = 0,
r 1 = 5 and r 2 = 25.
4.14 Customers asking for a certain product arrive according to a Poisson process with
rate λ. The demand sizes of the customers are independent random variables and have a
common discrete probability distribution {p k , k = 1, 2, . . . }. Any demand that cannot be
directly satisfied from stock on hand is back ordered. The control rule is based on the
inventory position, which is defined as the stock on hand minus the amount back ordered
plus the amount on order. Each time the inventory position reaches the reorder level s or
drops below it, the smallest multiple of the basic quantity Q is ordered to bring the inventory
position level above s. The lead time of any replenishment order is a fixed constant L >
0.
(a) Prove that the limiting distribution of the inventory position is a discrete uniform
distribution. (Hint: use relation (4.3.2) and verify that the one-step transition matrix of the
embedded Markov chain is doubly stochastic.)
(b) Derive the limiting distribution of the stock on hand.
(c) What is the average replenishment frequency and what is the average stock on hand?
(d) What is the fraction of customers whose demands are (partially) back ordered? What
is the fraction of demand that is not satisfied directly from stock on hand?
4.15 Consider the transient probabilities p ij (t) in a continuous-time Markov chain with finite
space I = {1, . . . , n}. Let the n × n matrix Q be defined as in the proof of Theorem 4.5.2.
Assume that the matrix Q has n different eigenvalues λ 1 , . . . , λ n . Let a k be an eigenvector
corresponding to the eigenvalue λ k for k = 1, . . . , n and let S be the n×n matrix whose kth
column vector is a k . For each initial state i, denote by p i (t) the vector whose jth element
equals p ij (t). Use results from Section 1.4 to verify the representation
n
λ k t
p i (t) = c ik e a k , t ≥ 0,
k=1
for constants c i1 , . . . , c in , where the vector c i = (c i1 , . . . , c in ) is given by c i = S −1 e i
with e i denoting the ith unit vector (0, . . . , 1, . . . , 0).
4.16 An operating system has r + s identical units where r units must be operating and s
units are in preoperation (warm standby). A unit in operation has a constant failure rate of
λ, while a unit in preoperation has a constant failure rate of β with β < λ. Failed units enter
a repair facility that is able to repair at most c units simultaneously. The repair of a failed
unit has an exponential distribution with mean 1/µ. An operating unit that fails is replaced
immediately by a unit from the warm standby if one is available. The operating system
goes down when less than r units are in operation. Show how to calculate the probability
distribution function of the time until the system goes down for the first time when all of
the r + s units are in good condition at time 0.
4.17 An electronic system uses one operating unit but has built-in redundancy in the form
of R standby units. The standby units are not switched on (cold standby). The operating
unit has an exponentially distributed lifetime with mean 1/λ. If the operating unit fails,
it is immediately replaced by a standby unit if available. Each failed unit enters repair
immediately and is again available after an exponentially distributed repair time with mean
1/µ. It is assumed that the mean repair time is much smaller than the mean lifetime. There
are ample repair facilities. The system is down when all R +1 units are in repair. Assuming
that all R + 1 units are in perfect condition at time 0, let the random variable τ be the time
until the first system failure.