Question

Regional Airlines is establishing a new telephone system for handling flight reservations. During the 10:00 AM to 2:00 PM time period, calls to the reservation agent occur randomly at an average rate of one call every 3.75 minutes. Historical service time data show that a reservation agent spends an average of 3 minutes with each customer. The waiting line model assumptions of Poisson arrivals and exponential service times appear reasonable for the telephone reservation system.
At a planning meeting, Regional’s management team agreed that an acceptable customer service goal is to answer at least 75% of the incoming calls immediately. During the planning meeting, Regional’s vice president of administration pointed out that the data show that the average service rate for an agent is faster than the average arrival rate of the telephone calls. The vice president’s conclusion was that personnel costs could be minimized by using one agent and that single agent must be able to handle the telephone reservations and still have some idle time. The vice president of marketing restated the importance of customer service and expressed support for at least two reservation agents.
We already analyzed the single-agent system’s performance according to the opinion of vice president of administration. We concluded that operating the telephone reservation service with only one ticket agent appears unacceptable because they won’t be able to meet their goal. Answer to the following questions to evaluate the opinion of vice president of marketing:

1. What is the service rate for the 2-agent system? Interpret the number.





2. What percentage of time both agents are idle?





3. What percentage of time a caller will be blocked if the system design does not allow callers to wait?



4. What percentage of time only one ticket agent is busy?

Answer 1

ANSWER::

Let         n            =            Number of callers (customers) in the system,

              pn           =            Probability of n callers (customers) in the system,

              c             =            Number of parallel ticket agents (service channels) in the system,

              λ            =            Arrival rate of customers, i.e. average number of calls arriving per hour,

              µ            =            Service rate of individual agent, average number of calls served per hour per agent

              ρ            =            Average utilization of the system

It is given that –

a.           Number of agents is 2, i.e.           c             =            2

b.           One call arrives every 3.75 minutes, i.e. Arrival rate –

              λ            =            60 / 3.75             =            16 calls per hour

c.           One call is served every 3 minutes, i.e. Service rate –

              µ            =            60 / 3                  =            20 calls per hour per agent

d.           Service goal is to answer at least 75% of the calls immediately. It means that the probability of less than 2 customers in the system should be more than 75%. It can be written as –

              p0           +            p1           ≥            0.75

Following is the solution of given questions:

1.           Service Rate for the 2-agent system:

Service rate per agent is µ = 20 calls per hour per agent. So, service rate for the 2-agent system is –

              =            c * µ      =            2 * 20    =            40 calls per hour              ANSWER

Interpretation: Doubling the number of agents doubles the average number of calls answered. To compare with arrival rate ( λ ) of 16 calls per hour, we can find average utilization of the system as follows –

ρ            =            λ / (c * µ)            =            16 / (2 * 20)       =              40%

Hence, on an average basis, this service rate looks pretty good. However, the Service goal probabilities should be analyzed as follows:

2.           Percentage of time both agents are idle:

Both agents are idle when there is zero customer (call) in the system. Hence, the percentage of time both agents are idle is equal to the probability of zero customers in the system, i.e. –

             

             

             

             p0           =            0.4286

So, both agents are idle for 42.86% of time.         ANSWER

3.           Percentage of time a caller is blocked, because system (both agents) is busy:

This can be derived from the probability that a customer has to wait, which is as follows:

             

             

             

             

So, a caller is blocked 22.86% of time, as the system is busy.        ANSWER

4.           Percentage of time only one agent is busy:

This means that there is only one customer in the system. So, we have to derive the probability of 1 call in the system, which is:

p1           =            1 - (Probability of zero customers in the system + Probability of two or more customers in the system)

p1           =            1 - (p0 + pn≥c)

Now, putting in these values from the part 2 and 3 as calculated above.

p1           =            1 – ( (3/7) + (8/35) )

p1           =            1 – ( 23/35 )

p1           =            12/35    =            0.3429

So, only one agent is busy 34.29% of the time.    ANSWER

NOTE:: I HOPE YOUR HAPPY WITH MY ANSWER....***PLEASE SUPPORT ME WITH YOUR RATING...

***PLEASE GIVE ME "LIKE"...ITS VERY IMPORTANT FOR ME NOW....PLEASE SUPPORT ME ....THANK YOU