Handbook of Civil Engineering Calculations

TABLE 37

Given Frequency of Relative frequency

outcome Successor successor of successor

A 5 5/12 = 0.4167

AB 4 4/12 = 0.3333

C J_ 3/12 = 0.2500

Total 12

A 5 5/10 = 0.5000

BB 3 3/10 = 0.3000

C _2 2/10 = 0.2000

Total 10

A 2 2/13 = 0.1538

CB 3 3/13 = 0.2308

C _8 8/13 = 0.6154

Total 13

4. Compute the expected number of units in use
   at steady-state conditions
   In the preceding calculation procedure, the total number of machines that will be in use si-
   multaneously is 1200. Multiply the steady-state probabilities found in step 3 by 1200 to
   obtain the expected number of units of each model that ultimately will be in use simulta-
   neously. The results are: model A, 1200(12/35) = 411; model B, 1200(10/35) = 343; mod-
   el C, 1200(13/35) = 446. These results coincide with those obtained in the preceding cal-
   culation procedure, and so the latter are confirmed.
   Related Calculations: In constructing the recurring series of outcomes, it is nec-
   essary to apply the principle of succession. Assume that a Markov process has three pos-
   sible outcomes, A, B, and C; and let Af(A-B) = number of times that A is followed by B.
   The principle is W(A-B) + N(A-C) = N(B-A) + N(C-A). As an illustration, consider the
   following skeletal series, where each outcome is followed by a different outcome:

A-C-A-C-B-A-B-A-C-B-A-C-B-C-A-C

The last outcome will be followed by A. Then AT(A-B) + N(A-C) = N(B-A) + N(C-A)
= 6; N(B-A) + TV(B-C) = TV(A-B) + N(C-B) = 4; N(C-A) + N(C-B) = N(A-C) + TV(B-C) =

6. Now this skeletal series can be expanded to the true series by allowing one outcome to
   be followed by the same outcome. For example, assume the requirements are TV(A-A) = 3,
   TV(B-B) = 5, and TV(C-C) = 6. A true recurring series is

A-A-C-C-C-C-A-C-C-C-B-B-B-A-B-B-A-C-C-B-B-B-A-A-C-B-C-A-A-C