Question

A vending machine dispenses hot chocolate or coffee. Service time is 20 seconds per cup and is constant. Customers arrive at a mean rate of 64 per hour, and this rate is Poisson-distributed.

Answer 1

Answer:

A vending machine dispenses hot chocolate or coffee. Service time is 20 seconds per cup and is constant. Customers arrive at a mean rate of 64 per hour, and this rate is Poisson-distributed. Determine:

a.Average number of customers waiting in line.

b.Average time customers spend  in the system.

The average number of customers waiting is 0.196 and average time spend is 0.00862 hr.

Step-by-step explanation:

Given:

Arrival rate, $\lambda$ = 64/hr

Service time = 20 sec/cup

Service rate $\mu$ = $\frac{60}{20}$ = $3$ per/min = $3\times 60$ = $180$ per/hr

According to the question:

Its a Poisson's distribution of (M/M/1) model.

Formula to be used:

a.Average number of customers waiting in line.

 ⇒  $L_q=\frac{\lambda^2}{\mu(\mu-\lambda)}$

b.Average time customers spend  in the system.

 ⇒ $W=\frac{1}{\mu-\lambda}$

Solving step-wise:

a.Average no. of customer waiting.

 ⇒ $L_q=\frac{\lambda^2}{\mu(\mu-\lambda)}$  

 ⇒ $L_q=\frac{64\times 64}{180(180-64)}$

 ⇒ $L_q=0.196$

b.Average time spend.

 ⇒ $W=\frac{1}{\mu-\lambda}$

 ⇒ $W=\frac{1}{180-64}$

 ⇒  $W = 0.00862$ hr

So the average number of customers waiting is 0.196 and average time spend is 0.00862 hr.