# Two stage cyclic queue

Given a cyclic queue of two servers of exponential service rates, if there are N customers at one server at time t, how do i start about showing that N can be modeled as a birth and death process? and then find the BD rates??

Totally have no idea how to begin. what are we supposed to show to prove it can be modeled as a BD process?

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 Assume that at time $t$ there are $N$ customers at server $A$ and $n-N$ at server $B$. What events can happen next (there are at most two)? How much time before these happen? What state $(N,n-N)$ of the process just after that time? – Did Apr 15 '12 at 8:11 events that can happen next will be that the arrival rate of server B will be service rate of server A and arrival rate of server A will be service rate of server B? So the time taken will be dependent on the arrival rate? State after tt time will be (n-N, N)? Still a bit confused by what it is asking for though. – edelweiss Apr 15 '12 at 10:04 These are not events (and you are merely restating hypotheses). So, you have N customers at server A and n-N at server B until either ____ and then ____ or ____ and then ____. – Did Apr 15 '12 at 11:22 hmm i dont really understand. if its a closed system, there will be N customers at server A and n-N at server B, so after that server A will be processing the N+1 customer and n-N+1 customer at server B. is this what you are referring to? – edelweiss Apr 15 '12 at 14:42 If server A ends serving a customer, this one goes to server B hence the state (N,n-N) is replaced by (N-1,n-N+1) (and not what you wrote). But server B could end serving one of its customers first, in which case (N,n-N) is replaced by _____ . That is, unless N=____ or N=____, in which case only ____ can happen. Now, at which rate do the transitions from (N,n-N) to (N+1,n-N-1) and to ____ occur? – Did Apr 15 '12 at 15:23