The Federal Emergency Management Agency (FEMA) has come in to help manage the di

Question

The Federal Emergency

Management Agency (FEMA) has come in to help manage the disaster

relief efforts after a recent hurricane. Horatio has been assigned as the initial contact for a

particular district when people call in seeking help. His job is to speak with each person w

ho

calls in and transfer them to the appropriate department (e.g., housing, food). Because the callers

need help, we will assume they will stay on the line

no matter

how many people are waiting.

We

also assume that the interarrival and service times are

exponentially distributed.

Suppose that on average 15 people call in per hour, and the Horatio spends 3 minutes talking

with each person before he transfers them.

a)

Using Kendall’s notation, what type of queueing system is this?

b)

What are

,

,

and

?

c)

D

raw the transition rate diagram

. Because there is no limit to the number of people who

can be on hold, it should end with an ellipsis “...”

d)

What percentage of time

are

there no callers waiting to talk with Horatio?

e)

How many callers are in the system on av

erage?

f)

What percentage of time is Horatio not on the phone with anyone?

g)

How long will the average person who calls in have to wait until he or she talks with

Horatio?

h)

How long does the average caller spend on the phone before they’re transferred

(includ

ing both the wait time and talking with Horatio)?

i)

Horatio’s supervisor has said that she would not like callers to wait on hold for more than

5 minutes. However, they don’t have the resources to hire a second person so she

recommends Horatio try to spend

less time on the phone. On average, how long should

Horatio spend on the phone with each caller to ensure the wait time is on average 5

minutes?

Explanation / Answer

So time of call should be less than 2.62 mins

Ans A Kendall's notation M/M/1 Comment Ans B Arrival rate A 15 Ans B Service rate S 20 60/3 =20 Ans B U Utilization ratio U=A/S 0.75000 <1 Length of queue diminishing Ans E Ls Expected number of people in system Ls A/(S-A) 3 Ans H Ws Avg waiting time in the system Ws 1/(S-A) 0.20000 hours 12.0000 mins Ans G Wq Avg waiting time in the Queue Wq A/S(S-A) 0.15000 hous 9.0000 mins Lq Avg no of customer waiting to be served Lq A^2/S(S-A) 2.2500 Ans D Po Prob of 0 units in system Po 1-(A/S) 0.25000 Ans F 25% Goal seeking with Wq =5 Ans i Wq 5.00039 Service rate (60/TS) 22.86997 TS Time spent 2.62353

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