11. In the simple queueing model of Example 11.6 suppose that the waiting line can be at most K individuals (including the one in service). Write the Kolmogorov Forward Equations and determine A. Example 11.7. A more general birth-and-death process If in the simple queuing process, the arrival rates and the service rats are different depending on the system states, then we have a more general birth-and-death process. More specifically, in this process, we have 2012, 22 =2 y = ... and 20 = 4,2 = 12, 132 = My, ... . Then we have 100 = 2,2=-(2 + 4), 12 =-(+1), ..., to make each row of the rate matrix to sum up to 0. The matrix then is: 0 0 2 (1+4) 0 A= 24 - (2+) 0 2 an the Kolmogorov differential equations will be Pield) = -2.po(t) + H, Pat) p.(0) = 1.-P.:-(1)-(a,+ l)p, (+ 4P(0,1 =0,1,2,..., j = 1,2,... 11. In the simple queueing model of Example 11.6 suppose that the waiting line can be at most K individuals (including the one in service). Write the Kolmogorov Forward Equations and determine A. Example 11.7. A more general birth-and-death process If in the simple queuing process, the arrival rates and the service rats are different depending on the system states, then we have a more general birth-and-death process. More specifically, in this process, we have 2012, 22 =2 y = ... and 20 = 4,2 = 12, 132 = My, ... . Then we have 100 = 2,2=-(2 + 4), 12 =-(+1), ..., to make each row of the rate matrix to sum up to 0. The matrix then is: 0 0 2 (1+4) 0 A= 24 - (2+) 0 2 an the Kolmogorov differential equations will be Pield) = -2.po(t) + H, Pat) p.(0) = 1.-P.:-(1)-(a,+ l)p, (+ 4P(0,1 =0,1,2,..., j = 1,2