Consider a single server queue with a Poisson arrival process at rate A, and exponentially distributed...

  

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Consider a single server queue with a Poisson arrival process at rate A, and exponentially distributed service times with rate u. All interarrival times and service times are independent of each other. This is similar to the standard M|M|1 queue, but in this queue, as the queue size increases, arrivals are more and more likely to decide not to join it. If an arrival finds n people already in the queue ahead of them (including anyone being served), then they join with probability 1/(n+1). Let N(t) be the number in the queue at time t. (a) Draw the transition diagram for this queue, including the transi- tion rates. (b) Write down equations that the equilibrium distribution satisfies. (c) Find the equilibrium distribution for this queue, when it exists. (d) What are conditions on A and under which the equilibrium distribution exists? Consider a single server queue with a Poisson arrival process at rate A, and exponentially distributed service times with rate u. All interarrival times and service times are independent of each other. This is similar to the standard M|M|1 queue, but in this queue, as the queue size increases, arrivals are more and more likely to decide not to join it. If an arrival finds n people already in the queue ahead of them (including anyone being served), then they join with probability 1/(n+1). Let N(t) be the number in the queue at time t. (a) Draw the transition diagram for this queue, including the transi- tion rates. (b) Write down equations that the equilibrium distribution satisfies. (c) Find the equilibrium distribution for this queue, when it exists. (d) What are conditions on A and under which the equilibrium distribution exists?