Also,^+^+^=1200 (d)- Solve the system of equations
The results are XAu = 411; Jf 5 M = 343; XCiU = 446. Thus, it is expected that there will ulti-
mately be 411 units of model A, 343 units of model B, and 446 units of model C in use si-
multaneously. Note that the values of XA>39 XE^ and A^ 3 in Table 36 are very close to the
limiting values. Thus, the expected values approach their respective limits rapidly.
Related Calculations: Many problems in engineering, economics, and other ar-
eas lend themselves to solution as Markov processes. The computational techniques ap-
plied in this calculation procedure and the preceding one are entirely general, and they
may be applied to any problem where a Markov process exists.
VERIFICATION OF STEADY-STATE
CONDITIONS FOR A MARKOV PROCESS
Verify the accuracy of the results obtained in the preceding calculation procedure by de-
vising an alternative method of solution.
Calculation Procedure:- Construct a recurring series of outcomes that conforms
with the given process
Assume that a Markov process has three possible outcomes, A, B, and C, and that the first
35 outcomes were these:
B-A-A-B-B-A-C-C-C-C-B-A-A-A-C-C-A-A-A-C-C-B-A-B-A-B-C-C-C-C-A-B-C-B-B
This series consists of 12 A's, 10 B's, and 13 Cs. Also assume that this series of out-
comes will recur indefinitely. Thus, the last outcome in the series will be followed by B.
It will be demonstrated that this series is relevant to the preceding calculation procedure.
- Compute the transition probabilities as established
by the recurring series
Count the successors of the outcomes in this series, and then compute the relative fre-
quencies of the various successions. Refer to Table 37 for the calculations. Since the giv-
en series of outcomes will recur indefinitely, the relative frequencies in Table 37 equal the
transition probabilities corresponding to the present Markov process. Thus, the probabil-
ity that A will be followed by B is 0.3333, and the probability that C will be followed by
A is 0.1538. Since these transition probabilities coincide with those in Table 35, it follows
that the present recurring series provides a basis for investigating the Markov process in
the preceding calculation procedure. - Compute the steady-state probabilities
In the long run, the probability that a given outcome will occur is independent of some
outcome in the distant past. In the recurring series, the relative frequencies of the out-
comes are: outcome A, 12/35; outcome B, 10/35; outcome C, 13/35. These relative fre-
quencies are the steady-state probabilities corresponding to the Markov process.