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Problem:

A service station has three servers, indexed 1, 2, and 3 . When a customer arrives, he is assigned to the idle server with the lowest index. If all servers are busy, the customer goes away. The service times at server $i$ are independent and exponentially distributed with mean $1 / \mu_i$, for $i=1,2,3$, where $\mu_i \neq \mu_j$ if $i \neq > j$. The customers arrive according to a Poisson process with rate $\lambda$. Define a continuous-time Markov chain to analyze the number of customers at the service station and specify its parameters.

Part of the solution:

Answer Let $Y(t)$ be the number of customers at the service station at time $t$. However, the process $\{Y(t), t \geq 0\}$ cannot be directly analyzed by a continuous-time Markov chain. In fact the sojourn time for state 2 is not exponentially distributed. Instead we investigate $Y(t)$ indirectly by considering another process first.

What I do not understand yet is my specifically the sojourn time of state 2 is not exponentially distributed. It would be extremely helpful if somebody could show me the intermediate steps to arrive to this conclusion.

Tim
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1 Answers1

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When $Y(t)=2$, you have 2 customers in the service station. However, you do not know which servers are busy and which servers are empty. Therefore, how will you determine the transition rates from that state to the other states? In order for the system to be Markovian, you need to have, at each state, all information required to determine the transition to the upcoming state. Therefore, you need to store, at each state, whether servers 1, 2 and 3 are busy. You need 3 state variables: B1, B2, B3, respectively. Of course, their sum is $Y$, i.e., $Y=B1+B2+B3$. However, from $Y$ you cannot determine $B1, B2$ and $B3$.

Daniel S.
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