A familiar sight in Middle East street scenes are the so-called sheroots. A sher

Question

A familiar sight in Middle East street scenes are the so-called sheroots. A sheroot is a seven-seat cab that drives from a fixed stand in a town to another town. A sheroot leaves as soon as all seven seats are occupied by passengers. Consider a sheroot stand which has room for only one sheroot. Potential passengers arrive at the stand according to a Poisson process at rate lambda. If upon arrival a potential customer finds no sheroot present and seven other customers already waiting, the customer goes elsewhere for transport; otherwise, the customer waits until a sheroot departs. After a sheroot leaves the stand, it takes an exponential time with mean 1/mu until a new sheroot becomes available. Formulate a continuous-time Markov chain model for the situation at the sheroot stand. Specify the state variable(s) and the transition rate diagram.

Explanation / Answer

X1(t)=number of waiting passangers at t

X2(t)=1(sheroot is present at time t) or 0(otherwise).

state space I={(i,0):i=0,1....7} U {(i,1):i=0,1,2....6}.

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