An M/M/1 queue at a coffee shop has an average service time of 1.25 minutes, and

Question

An M/M/1 queue at a coffee shop has an average service time of 1.25 minutes, and an

arrival rate of 36 customers per hour.

(a) What is the average number of customers in the system?

(b) What percentage of customers have a time in system less than 5 minutes?

You join this queue when there are 4 customers in the system ahead of you (3 in the queue,

and 1 in service).

(c) If your time in system is t minutes, what is the probability density function for t ?

(d) What is the probability that your time in system t is less than 5 minutes?

An M/M/i queue a coffee shop has an average service time of 1.25 minutes, and an and arrival rate of 36 customers per hour. (a) What is the average number of customers in the system? (b) What percentage of customers have a time in system less than 5 minutes? You join this queue when there are 4 customers in the system ahead of you (3 in the queue, and 1 in service (c) If your time in system is t minutes, what is the probability density function for t (d) What is the probability that your time in system t is less than 5 minutes?

Explanation / Answer

= 36 customers/hour

µ= 60/1.25 = 48 customers /hours

The question follows M/M/1 : / FIFO model

a)Average number of customers in the system : Ls

Ls = /(µ - ) = 36/(48-36) = 3

b)Percentage of customers have a time in system less than 5 minutes

P(N < 5) = 1 - P( N > 5) = 1 – (/µ)5+1 =1- 0.17797 = 0.82203

Percentage of customers have a time in system less than 5 minutes = 82.2 %

c)the probability density function of the waiting time in the system is given :

f(t) = (µ -)e(µ-)t

d)probability that time in system t is less than 5 minutes:

P(ws < 5 ) = 1 – P(ws >5) = 1 – e-(µ-)5 = 1 – e-60 = 0.99

Note: Go through the formula for the given model.

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