Dr. Tarun Gupta, a Michigan vet, is running a rabies vaccination clinic for dogs

Question

Dr. Tarun Gupta, a Michigan vet, is running a rabies vaccination clinic for dogs at the local grade school. Tarun can shoot a dog every 2 minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of one dog every 6 minutes according to a Poisson distribution. Also assume that Tarun's shooting times are exponentially distributed. Compute the following:

a) The probability that Tarun is idle.

b) The proportion of the time that Tarun is busy.

c) The average number of dogs being vaccinated and waiting to be vaccinated.

d) The average number of dogs waiting to be vaccinated.

e) The average time a dog waits before getting vaccinated.

f) The average amount of time a dog spends waiting in line and being vaccinated. Please show steps.

Explanation / Answer

arrival rate=L=1 per 6 min = 1/6 per min=0.167 dog per min

Service rate=m= 1 per 2 min= 1/2 per min =0.5 dog per min

Hence L/m=0.167/0.5=0.334

a) The probability that Tarun is idle.= 1- L/m=1-0.334=0.666

b) The proportion of the time that Tarun is busy.=L/m=0.334

c) The average number of dogs being vaccinated and waiting to be vaccinated=Lq+L/m=L^2/(m*(m-L) +L/m=0.167^2/(0.5*(0.5-0.167))+0.334=0.5015

d) The average number of dogs waiting to be vaccinated.=Lq=L^2/(m*(m-L)=0.167^2/(0.5*(0.5-0.167))=0.167

e) The average time a dog waits before getting vaccinated.=Lq/L=0.167/0.167=1

f) The average amount of time a dog spends waiting in line and being vaccinated= Lq/L +1/m=1+1/0.5=.1+2=3

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