Question

A salesman territory consists of three cities A,B and C. He never sells in

the same city on successive days. If he sells in A, then the next day he

sells in city B. However if he sells in either B or C, the next day he is

twice as likely to sell in the city A as in the other city. In the long run,

how often does he sell in each of the cities?

Answer 1

Step-by-step explanation:

I hope this will help you

Answer 2

Answer:

In the long run, the salesman sells 40% in city A, 45% in city B, and 15% in city C.

Step-by-step explanation:

Step 1:

The TPM of the Markov chain is

$P=\left[\begin{array}{ccc}0&1&0\\\frac{2}{3} &0&\frac{1}{3} \\\frac{2}{3} &\frac{1}{3} &0\end{array}\right]$

where the first row and column are for city A, the second row and column are for city B, and the third row and column are for city C.

Step 2:

Let π = (π1, π2, π3) be the steady-state distribution of the chain.

We know that,

πP = π ...(i)

π1 + π2 + π3 = 1 ...(ii)

Step 3:

From equation (i), we get

(π1, π2, π3) $\left[\begin{array}{ccc}0&1&0\\\frac{2}{3} &0&\frac{1}{3} \\\frac{2}{3} &\frac{1}{3} &0\end{array}\right]$ = (π1, π2, π3)

Using matrix multiplication

$(\frac{2}{3})$(π2) + $(\frac{2}{3})$(π3) = π1 ...(iii)

π1 + $(\frac{1}{3} )$(π3) = π2 ...(iv)

$(\frac{1}{3})$(π2) = π3 ...(v)

Step 4:

Substituting equation (v) in equation (iii)

$(\frac{2}{3})$(π2) + $(\frac{2}{3})(\frac{1}{3})$(π2) = π1

$(\frac{6+2}{9} )$(π2) = (π1)

π1 = $(\frac{8}{9})$(π2) ...(vi)

Substituting π1 and π3 in equation (ii)

$(\frac{8}{9})$(π2) + π2 + $(\frac{1}{3})$ (π2) =  1

 π2 = $\frac{9}{20}$

Substituting π2 in equation (vi), we get

π1 = $(\frac{8}{9})(\frac{9}{20})$

π1 = $\frac{8}{20}$

Substituting π2 in equation (v), we get

π3 = $(\frac{1}{3})(\frac{9}{20})$

π3 = $\frac{3}{20}$

Therefore, the steady state distribution of the chain is

π = $(\frac{8}{20} \frac{9}{20} \frac{3}{20} )$

Step 4:

The percentage value is

π = $([\frac{8}{20}*100] [\frac{9}{20}*100] [\frac{3}{20}*100] )$

π = (40%  45%  15%)

Therefore, in the long run, the salesman sells 40% of his item in city A, 45% of his items in city B, and 15% of his items in city C.

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