Question

Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can vaccinate a dog every 3 minutes. It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of 1 dog every 4 minutes, according to a Poisson distribution. Also assume that Sam's vaccinating times are exponentially distributed. Find the:
(a) Probability that Sam is idle.
(b) Proportion of time that Sam is busy.
(c) Average number of dogs receiving or waiting to be vaccinated.
(d) Average number of dogs waiting to be vaccinated.
(e) Average time a dog waits before getting vaccinated.
(f) Average amount (mean) of time a dog spends between waiting in line and getting
vaccinated.

Answer 1

SOLVE ANSWERS

Explanation:

Given:

Arrival rate  = 1 Dog per 4 minute  = 1/4 = 0.25

Vaccinate rate = 1 Dog per 3 minute  = 1 / 3 = 0.33

Computation:

A. Probability that Sam is idle = 1 - [(Arrival rate ) / (Vaccinate rate)]

Probability that Sam is idle = 1 - (0.25 / 0.33)

Probability that Sam is idle = 0.25

B. Proportion of the time Sam is busy = [(Arrival rate ) / (Vaccinate rate)]

Proportion of the time Sam is busy = (0.25 / 0.33)

Proportion of the time Sam is busy = 0.75

C. Average number of dogs receiving or waiting to be vaccinated = Arrival rate / ( Vaccinate rate - Arrival rate )

Average number of dogs receiving or waiting to be vaccinated = 0.25 / (0.33 - 0.25)

Average number of dogs receiving or waiting to be vaccinated = 3.125

E . Average time a dog waits before getting vaccinated = [(Arrival rate ) / (Vaccinate rate)] [1/(Vaccinate rate / Arrival rate)]

Average time a dog waits before getting vaccinated = [0.25 / 0.33][1 / (0.33 /0.25)]

Average time a dog waits before getting vaccinated = 1

F. Average amount (mean) of time a dog spends between waiting in line and getting vaccinated = 1 / (Vaccinate rate - Arrival rate)

Average amount (mean) of time a dog spends between waiting in line and getting vaccinated = 1 / (0.33 - 0.25)

Average amount (mean) of time a dog spends between waiting in line and getting vaccinated = 12.5 min

Answer 2

Answer with Explanation:

Since we have given that

The dogs will arrive independently and randomly throughout the day at a rate of 1 dog every 4 minutes.

Sam vaccinate a dog every 3 minutes i.e. $\lambda=\dfrac{1}{3}$

a) Probability that Sam is idle = $\dfrac{1}{4}=0.25$

b) Proportion of time that Sam is busy =

$\rho=\dfrac{\lambda}{\mu}=\dfrac{\dfrac{1}{4}}{\dfrac{1}{3}}=\dfrac{3}{4}=0.75$

c) Average number of dogs receiving or waiting to be vaccinated is given by

$L=\dfrac{\rho }{1-\rho }=\dfrac{0.75}{1-0.75}=3$

d)  Average number of dogs waiting to be vaccinated is given by

$L_q=\dfrac{\rho^2 }{1-\rho}=\dfrac{0.75^2}{1-0.75}=2.25$

e) Average time a dog waits before getting vaccinated is given by

$W_q=\dfrac{L_q}{\lambda}=\dfrac{2.25}{0.25}=9\ minutes=0.15\ hour$

f) Average amount (mean) of time a dog spends between waiting in line and getting vaccinated is given by

$\dfrac{1}{\mu-\lambda}=\dfrac{12}{1}=12=0.2\ hour$