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 3 minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of one dog every 5 minutes according to a Poisson distribution. Also assume that​ Tarun's shooting times are negative exponentially distributed. ​

a) The probability that Tarun is idle​ = . 4 ​(round your response to two decimal​ places). ​
b) The proportion of the time that Tarun is busy​ = . 6 ​(round your response to two decimal​ places). ​
c) The average number of dogs being vaccinated and waiting to be vaccinated​ = 1.5 dogs​ (round your response to two decimal​ places). ​
d) The average number of dogs waiting to be vaccinated​ = . 9 dogs​ (round your response to two decimal​ places). ​
e) The average time a dog waits before getting vaccinated​ = nothing minutes​ (round your response to two decimal​ places).

Answer 1

a) Probability that Tarun is idle = 0.4

b) Proportion of the time that Tarun is busy = 0.6

c) The average number of dogs being vaccinated and waiting to be vaccinated​ = 1.5 dogs

d) The average number of dogs waiting to be vaccinated​ = 0.9 dogs

e) The average time a dog waits before getting vaccinated​ = 0.075 mins = 4.5 hrs

Explanation:

a) Let the probability that Tarun is idle = P

Probability that Tarun is not idle = P₀

P + P₀ = 1

Average number of dogs shot in 1 hr(60 mins),
`\mu = 60/3\\` = 20 dogs/hr

Rate of dog arrival,
`\lambda = 60/5` = 12 dogs/hr

Probability that Tarun is idle, P = 1 - P₀

`P = 1 - (\lambda)/(\mu) \\P = 1 - (12)/(20) \\P = 1 - 0.6`

P = 0.4

b) Proportion of the time that Tarun is busy(not idle):

`P_(0) = (\lambda)/(\mu) \\P_(0) =(12)/(20)`

`P_(0) =0.6`

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

`n = (\lambda)/(\mu - \lambda) \\n = (12)/(20 - 12) \\n = 12/8`

n = 1.5 dogs

d) The average number of drugs waiting to be vaccinated:

`n = (\lambda^(2) )/(\mu ( \mu - \lambda))\\ n = (12^(2) )/(20 ( 20 - 12))`

n = 0.9 dogs

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

`Wating time = (\lambda)/(\mu ( \mu - \lambda)) \\Waiting time = (12)/(20 (20 - 12))`

Waiting time = 0.075 minutes

Waiting time = 0.075 * 60

Waiting time = 4.50 hrs