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A service center consists of two servers, each working at an exponential rate of two services per hour. If customers arrive at a Poisson rate of three per hour, then, assuming a system capacity of at most three customers, What fraction of potential customers enter the system?

I was thinking if I could think of the servers as one big system since the customers don't have a preference one way or another.

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2 Answers 2

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Time between arrivals is exponential with mean $1/\lambda=1/3$ hour. Service rate $\mu=2$ per hour. Length of a service time is exponential with mean $1/\mu=0.5$ hours.

This is an $M/M/2/3$ queue. 2 servers with $N=3$ max capacity which means a waiting room of size $1.$

You can use the total number currently in the system as the state space (as opposed to a vector keeping track of which server is busy.) This gives:

state $= 0$ means no one in system.

state $= 1$ means 1 customer being served, no one waiting

state $= 2$ means 2 customers being served, no one waiting

state $= 3$ means 2 customers being served, 1 waiting.

no other states possible.

We want the stationary distribution. It is a birth and death process so we know the answer from the general case. Or we can use the "balance equations." (Tricky part: departure rate in a state is proportional to the number of busy servers.)

After some algebra, $\pi_0=32/143$ and $\pi_3=27/143.$ Check my work.

Now the probability that an arrival is turned away since the system is full is $\pi_3.$ So then we know the proportion of arrivals that are able to get into the system. (Technical point: this is because the PASTA property holds here.)

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fraction of potential customers enter the system = 1 - πN
here N is system capacity = 1 - π3
= 1-27/143
= 0.811188

similar question with solution {see question 3}
https://www2.isye.gatech.edu/~ashapiro/publications/MT2-solutions.pdf

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