Respuesta :
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, [tex]\lambda[/tex] = 64/hr
Service time = 20 sec/cup
Service rate [tex]\mu[/tex] = [tex]\frac{60}{20}[/tex] = [tex]3[/tex] per/min = [tex]3\times 60[/tex] = [tex]180[/tex] 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.
⇒ [tex]L_q=\frac{\lambda^2}{\mu(\mu-\lambda)}[/tex]
b.Average time customers spend in the system.
⇒ [tex]W=\frac{1}{\mu-\lambda}[/tex]
Solving step-wise:
a.Average no. of customer waiting.
⇒ [tex]L_q=\frac{\lambda^2}{\mu(\mu-\lambda)}[/tex]
⇒ [tex]L_q=\frac{64\times 64}{180(180-64)}[/tex]
⇒ [tex]L_q=0.196[/tex]
b.Average time spend.
⇒ [tex]W=\frac{1}{\mu-\lambda}[/tex]
⇒ [tex]W=\frac{1}{180-64}[/tex]
⇒ [tex]W = 0.00862[/tex] hr
So the average number of customers waiting is 0.196 and average time spend is 0.00862 hr.
