Question

This is a reposted, pls do not copy the previous workings as the answer is wrong.

i.e.

Mean arrival rate = 1 customer/10mins = 6 customers/hr

Mean service time, E[Ts]=15mins

Mean service rate = 1/15 customer/min = 4 customers/hr

Traffic Intensity = Mean arrival rate / Mean service rate = 6/4 = 1.5 > 1

Q1. Arrival of customers to a local store may be modeled by a Poisson process with an average of 1 arrival every 10 min period. Each customer on average stays for a time exponentially distributed with mean 15 minutes. This is modeled as a birth-death process.

(i) What assumption is necessary for this process to be modeled as a birth-death process?

(ii) Compute the probability that the number of customers in the store reaches 30, if we have 3 customers at the start.

Pls explain with workings. Thxs

Answer 1

(i)For the process to be modeled as a birth-death ratio, the arrival and the service of the total number of customers during the specified time is the assumption that is necessary.

(ii)Since the number of customers at the start is three,

The arrival time =0

The service time=15 min

As the fourth customer arrives

The arrival time changes to 0+10

The service time =15+15

As the fifth customer arrives,

The arrival time=10+10

The service time =3×15

Continuing like this

As the nth customer arrives,

The arrival time =(n-3)×10

The service time=(n-2)×15

Therefore when n=30

The arrival time=(30-3)10=27×10=270 minutes=4 hours 30 minutes

The service time=(30-2)×15=28×15=420 minutes=7 hours

Probability =270/420=9/17<1