FIGURE 37. Replacement diagram
LONG-TERM FORECASTING
WITH A MARKOV PROCESS
With reference to the preceding calculation procedure, estimate the number of units of
each model that will ultimately be in use simultaneously.
Calculation Procedure:
- Form a system of simultaneous equations containing the limits
of the expected values
In the long run, the probability that a firm will buy a given model is solely a function of
the specific needs of that firm and the characteristics of each model; it is independent of
the particular model that the firm happens to own at present. Therefore, XA „, XB „, and
XCn approach definite limits as n increases beyond bound. The values of these variables
when n has a finite value constitute transient conditions, and the values when n is infinite
constitute the steady-state conditions. In practice, however, the steady-state conditions
may be considered to exist when all differences between transient and steady-state values
become less than some specified small number.
Let XA n = Hm XA n. In Eqs. a and b of the preceding calculation procedure, replace the
transient values with their respective limits and rearrange:
-0.5833^M + 0.500(LY^ + 0.1538^ = O (ar)
0.3333AT^M - 0.7000^u + 0.2308^Cw = O (b')
Present 1 year
hence
2 years
hence
3 years
hence
