Page 158 - A First Course In Stochastic Models

THE FLOW RATE EQUATION METHOD 151

In mathematical terms,

p j q jk = p k q kj .
j∈A k /∈A k /∈A j∈A
The balance principle (4.2.9) enables us to write down a recursive equation for the
p j when
I = {0, 1, . . . , N} and q ij = 0 for i ≥ 1 and j ≤ i − 2,

where N ≤ ∞. Then, by taking A = {i, . . . , N} with i = 0 and applying the
balance principle (4.2.9), we get

i−1 N

q i,i−1 p i = p k q kj , i = 1, . . . , N. (4.2.10)
k=0 j=i
This recursive relation is used quite often in queueing applications; see Chapter 5.
In queueing applications it is often the case that direct transitions from any state i
are either to higher states or to the state i − 1 directly below state i. A recursive
computation of the state probabilities is usually much faster than a computation by
any other method. Also the recursion scheme (4.2.10) is numerically stable since
it involves no subtractions.
Next we apply the flow rate equation method to the two examples discussed in
the previous section.

Example 4.1.1 (continued) Inventory control for an inflammable product
In this example the equilibrium probability p j represents the long-run fraction of
time that the stock in the tank equals j units. Assumptions 4.1.2 and 4.2.1 are
trivially satisfied (e.g. take state Q as regeneration state r). Using the transition
rate diagram in Figure 4.1.1 and equating the rate at which the process leaves state
i to the rate at which the process enters state i, it follows that

µp 0 = λp 1 ,
λp j = λp j+1 , j = 1, 2, . . . , Q − 1,
λp Q = µp 0 .

These equilibrium equations together with the equation p 0 + p 1 + · · · + p Q = 1
have a unique solution (in this special case an explicit solution can be given:
p 0 = (1 + Qµ/λ) −1 and p 1 = · · · = p Q = (µ/λ)p 0 ). Next we can answer the
questions posed earlier:
Q

the long-run average stock on hand = jp j (4.2.11)
j=0
the long-run fraction of demand that is lost = p 0 . (4.2.12)