Question

(20 pts) Inquiries arrive at an information center according to a Poisson process of rate 2 inquiries per second. It takes a server an exponential amount of time with mean ½ second to answer each query.

(a) (5 pts) If there are two servers. What is the probability that an arriving customer must wait to be served?

(b) (5 pts) Find the mean number of customers in the system and the mean time spent in the system.

(c) (10 pts) If the arrival rate increases to 10 inquiries per second and the number of servers is increased to 6, what is the resulting probability that an arriving customer finds all servers busy? What is the mean total delay for each inquiry? What is the percentage of all queries with waiting time less than 1 second?

Answer 1

Solution:

Hence,

a)

Given:

Arriving customer has to wait for means both the servers are busy, meaning that there are 2 customers in the system occupying both the servers, hence,

Using the above formula, we have,

Hence,

b)

Using the formula given above for M/M/k systems, we find the value of L to be,

The average number of customers in the system is 1.3333

Hence,

The average time spent in the system is 0.0111 minutes.

c)

New values,

Using the formula given above,

Hence,

We calculate the L using the above formula,

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