Assignment 2 Solution
ISyE 3044
1. A tool crib has exponential interarrival and service times and serves a very large group of mechanics.
The mean time between arrivals is 4 minutes. It takes 3 minutes on the average for a tool-crib attendant to
service a mechanic. The attendant is paid $10 per hour and the mechanic is paid $15 per hour. Would it be
advisable to have a second tool-crib attendant?
Solution:
The tool crib is modeled by an M/M/c queue (λ = 1/4, µ = 1/3, c = 1 or 2. Given that attendants are paid
$10 per hour and mechanics are paid $15 per hour,
Mean cost per hour is $10c + $15L. Assuming that mechanics impose cost on the system while in the queue
and in service.
CASE1: one attendant-M/M/1 (c = 1, ρ = λ/µ = 0.75)
L = ρ/(1 − ρ) = 3 mechanics
Mean cost per hour=$10 × 1 + $15 × 3 = $55 per hour.
Case2: two attendants-M/M/2 (c = 2, ρ = λ/cµ = 0.375)
L = cρ + [(cρ)c+1 P0 ]/[c(c!)(1 − ρ)2 ] = 0.8727
where
P0 = {[
c−1
X
(cρ)n /n!] + [(cρ)c (1/c!)(1/(1 − ρ))]}−1 = 0.4545
n=0
Mean cost perhour = $10 × +$15 × 0.8727 = $33.09 per hour.
Hence, it would be advisable to have a second attendant because long run costs are reduced by $21.91 per
hour.
2. A repair and inspection facility consists of two stations: a repair station with two technicians, and
an inspector station with 1 inspector. Each repair technician works at the rate of 3 items per hour; the
inspector can inspect 8 items per hour. Approximately 10% of all items fail inspection and are sent back
to the repair station. (This percentage holds even for items that have been repaired two or more times.) If
items arrive at the rate of 5 per hour, what is the long-run expected delay that items experience at each of
the two stations, assuming a Poisson arrival process and exponentially distributed service times? What is
the maximum arrival rate that the system can handle without adding personnel?
Solution:
1
x=
λ
= 5/0.9 = 5.556/hr
1 − 0.1
At the repair station (M/M/2):
ω=
1
1
=
= 2.34 hr
2
µ(1 − ρ2 )
3(1 − ( 5.556
2×3 ) )
At the inspection station(M/M/1):
ω=
1
= 0.41 hr
8 × (1 − 5.556/8)
The maximum arrival rate the system can handle without adding personnel is: λ = 2 × 3 × 0.9 = 5.4 hr
because the utilization at the repair stations are much higher than that at the inspection station, which
indicates the repair stations are the bottleneck of the system.
3. Call centers provide a convenient way for customers to get help with product problems. Cross-training
of workers is currently being considered in a number of industries as a way to improve productivity, reduce
the size of the workforce, and increase job satisfaction for workers. In the following, you are asked to evaluate
the redesign of a customer call center with cross-trained workers.
MySoft, Inc. develops software products in three basic areas: financial, personal productivity and emailers. They currently have a customer support call center that handles technical questions for owners of
their software from the hours of 8 AM to 4 PM Eastern Time. When a customer calls 888-555-Soft, they
listen to a recording that asks them to select among the three product lines. The number of customers
who can be connected at any one time is essentially unlimited. Each product line has its own operators. If
an appropriate operator is available, then the call is immediately routed to the operator; if an appropriate
operator is not available, then the caller is placed in a hold queue (and listens to a combination of music
and ads). Most questions are answered completely by the operators, but a few (4%) also have to be referred
to another technical group that prepares a response (two servers who can each work on one problem, firstcome-first-served). The customer does not stay on the line while this group works, but rather receives a
return phone call when the problem is solved. The return call is from someone in the same product-operator
group (but not necessarily the same operator).
MySoft, Inc. is hoping to reduce the total number of operators they need by cross-training operators so
that they can answer calls for any product line. Since the operators will not be experts across all products,
this is expected to increase the time to answer a question by about 10%. The question that MySoft, Inc.
has asked you to answer is how many cross-trained operators are needed to provide service at the same level
as the current system.
Models for the uncertain quantities in the system have been developed by another engineering group,
and are given below:
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(a) Arrival to the system follows a Poisson process with 1 call/min.
(b) Service time of operators follows an exponential distribution with 0.2 call/min.
(c) Service time of technicians follows an exponential distribution with 0.033 call/min.
Type of calls: financial 25%, productivity 34%, e-mailers 41%.
Current number of operators: financial 2, productivity 2, e-mailers 3.
Compute the approximate utilization of the current and the proposed systems if the number of operators
is not changed. Does it appear that a reduction in number of operators is possible?
Solution:
The delay of calls to choose operators does not affect the arrival rate or service rate of each call type thus
we can compute the arrival rate to each station from outside by applying decomposition property of Poisson
arrivals.

λ1 = 0.25 + 0.25λ4 
λ2 = 0.34 + 0.34λ4
λ + λ2 + λ3 − λ4 = 1
 1
λ3 = 0.41 + 0.41λ4
λ4 = 0.04(λ1 + λ2 + λ3 ) ⇒ λ4 = 0.04(1 + λ4 )
3
⇒ λ1 = 0.260, λ2 = 0.354, λ3 = 0.427, λ4 = 0.042
Service time of operators ∼ exponential with 0.2/1.1 = 0.182 min/call .
λ1 = 1 + λ4
λ4 = 0.04λ1
⇒ λ1 = 1.042, λ4 = 0.042
Technician process has the same arrival rate and service rate so Lq and Wq at the Technician process are
the same as before.
Comparison:
Sum of Lq for Financial,Productivity and Emailer of the current system= 8.604; average of Wq for Financial,
Productivity and Emailer of the current system= 8.223. Thus, the proposed system has a smaller expected
number of calls in the hold queue of 2.355 compared to 8.604 of the current system. When the number of
operators is reduced to 6 in the proposed system, Lq increases dramatically to 18.268; therefore, to provide
service at the same level as the current system, a reduction of the number of operators does not appear to be
possible. However, the proposed system with 7 operators will deliver higher customer-satisfaction in terms
of waiting time.
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