A family restaurant has one person processing the orders. On average, each custo

Question

A family restaurant has one person processing the orders. On average, each customer that comes to restaurant can be seated at the rate of 5 per minute. According to historical data, customers arrive at the rate of 15 per hour. Assume arrivals follow the Poisson distribution and service times follow the exponential distribution.

a. What is the average number of customers waiting in line?

b. What is the average time a customer spends in the waiting line?

c. What is the average number of customers in the system?

d. What is a customer's average time in the system?

e. The restaurant estimates that the arrival rate will increase to 10 per minute. Based on the simulation output for a two-server and a three-server system below, how many server would you suggest? Note that cost of waiting per minute is $13/customer and servers are paid $11/hr.

Number of channels 2 3 Arrival Rate 10 10 Service Rate 6 6 Probability of No Units in System .0909 .1727 Average Waiting Time .3788 .0375 Average Time in Sytem .5455 .2041 Average Number Waiting 3.7879 .3747 Average Number in System 5.4545 2.0414 Probability of waiting .7576 .2998 Probability of 11in System .0245 less than .0088

Explanation / Answer

Solution

Parts (a) to (d) pertain to M/M/1 queue system.

Given, average arrival rate per hour, = 15; average service rate per hour, µ = 300

[Note given is 5 per minute => (5 x 60) per hour.]

Terminology:

Number of customers waiting = m

Number of customers in the system = n

Waiting time = w

Total time spent in the system = v

Part (a)

Average number of customers waiting = E(m) = (2)/{µ(µ - )}= 152/{300(300 - 15)}

= 1/95 ANSWER

Part (b)

Average time in waiting = E(w) = ()/{µ(µ - )} = 15/{300(300 - 15)} = 1/1425 hour

= 2.5 seconds ANSWER

Part (c)

Average number of customers in the system = E(n) = ()/(µ - ) = 15/(300 - 15)

= 1/19 ANSWER

Part (d)

Average time spent in the system = E(v) = {1/(µ - )} = 1/(300 - 15) = 1/285 hour

= 12.6 seconds ANSWER

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