Chapter 14
Waiting Lines and
Queuing Theory Models
A Starting Example
There are on average 12 customers coming to
a candy store per hour. A cashier can take
care of a customer in 4 minutes on average.
At least how many cashiers should this store
hire if it does not want the customers
waiting for more than five minutes on
average?
Starting example (cont.)
• If the customers came in exactly every five
minutes, and service time is 4 minutes
exactly per customer:
• If the customers come randomly at the rate
of 12 per hour, and the service times are
random around 4 minutes:
Waiting Line Models
• Provide an analytical tool for the managers
to consider the trade-offs between the
customer satisfaction (in terms of customer
waiting time) and the service cost, if
customer arrivals and/or service times are
uncertain (random).
A Queuing System:
• is composed of customers, servers, and
waiting lines.
• A customer comes. If a server is idle, the
customer can be served immediately,
otherwise he/she has to wait in line.
Arrival Patterns
• Random arrival – arrivals follow Poisson
distribution.
– parameter:  , arrival rate (number of
customers per unit time)
• Scheduled arrivals -
Patterns of Service Time
• Random service time - The length of
service time follows the exponential
distribution.
– parameter:  ,service rate (number of
customers that can be served per unit time)
• Fixed service time -
Service Time and Service Rate
• Service rate 
= 1 / avg. service time on a customer
Characteristics of a Queuing
System:
•
•
•
•
Customer population – finite or infinite
Number of lines.
Number of service channels.
Number of service phases - number of
steps to finish a service.
• Priority rule - FIFO, LIFO, preemptive, ...
• Customer behavior – enter and stay, balk,
renege
Queuing Models in This Chapter
arrival
pattern
M/M/1
M/M/s
M/M/1
finite
service
time
pattern
random random
random random
random random
number
of
servers
1
s
1
population
infinite
infinite
finite
number
of
phases
1
1
1
priority
rule
customer
behavior
FIFO
no
balk,
no
renege
FIFO
no
balk,
no
renege
FIFO
no
balk,
no
renege
Performance of a Service System
Is Measured by:
• Average queue length (Lq) - average number of
customers in the waiting line.
• Average number of customers in the system (L).
• Average waiting time in the queue (Wq).
• Average staying time in the system (W).
• Utilization rate of servers (  ).
• Probability that n customers in the system (Pn).
‘In system’ vs. ‘In queue’
• ‘System’ contains ‘queue’ and service
facilities.
• ‘Number of customers in system’ counts
customers waiting in queue and customers
being served.
• ‘Number of customers in queue’ counts
customers waiting in queue only.
• Difference between ‘waiting time in system’
and ‘waiting time in queue’ - ?
Queuing System Calculations
• Use the formulas on p.601 (if doing handcalculations)
• Use QM for Windows (We use this
method!).
Requirements for Managerial
Users
• Using the calculation results of QM to
(1) analyze the performance of a service
system,
(2) make decisions on capacity such as
number of servers to hire.
M/M/m model
• Random arrivals, random service times, m
servers.
• Performance of an M/M/m queuing system
is determined by arrival rate , service rate
, and number of servers m.
• Software QM for Windows calculates the
performances of an M/M/S system. (Note:
use a same time base for both  and .)
Example: Arnold’s
Muffler Shop
(p.596)
• Time to install a new muffler is random, and
on average, the mechanic Reid Blank can
install 3 muffler per hour.
• Customer arrivals are random and at the rate
of 2 customers per hour on average.
• Evaluate this service system.
Questions about a Service
System
•
•
•
•
•
•
•
Probability of zero customer in system?
Utilization of the service capacity?
Avg. number of customers in system?
Avg. number of customers in line?
Average time a customer spends in system?
A customer’s average waiting time in line?
In what percent of time is the server idle?
Cost of a Service System
• Total cost
= Total service cost + Total waiting cost
• Total service cost
= (number of servers)·(unit labor cost)
• Total waiting cost =
(1) ·W·(unit waiting cost in system), or
(2) ·Wq· (unit waiting cost in queue).
Muffler Shop (2) p.598
• Waiting cost for the shop is $10 per hour
waiting in line.
• The mechanic Reid Blank is paid $7/hour.
• What is the total hourly cost of this system?
• What is the total daily cost of this system?
Muffler Shop (3) p.599
• If Jimmy Smith is hired to replace Reid
Blank, then the service rate can be
improved to 4 cars per hour, but Jimmy’s
hourly salary is $9.
• Evaluate the system with Jimmy Smith.
• Calculate the total daily cost of the system.
• Should Jimmy be hired to replace Reid?
Muffler Shop (4) p.602
• Suppose the shop opens a second garage
bay for installing mufflers, and a new
mechanic is hired whose salary and service
rate are same as Reid Blank.
• Evaluate the new system with two bays and
Reid Blank and the new mechanics.
• Calculate the total daily cost of the system.
• Is this a good alternative?
M/D/1 Model
• Random arrivals, fixed service time,
one server.
M/D/1 Example:
Compactor p.606
• A new compacting machine compacts a
truck of recycling cans in 5 minutes
constantly. Trucks coming randomly with
rate 8 trucks per hour.
• Evaluate this service system.
Compactor (2) p.606
• Cost for a truck waiting in queue is $60 per
hour.
• The amortized cost of the new compactor is
$3 per truck unloaded.
• Calculate the total cost for a truck unloaded.
• If the current truck waiting time in line is 15
minutes, then should the company purchase
the new compactor?
M/M/1
with Finite Population (Source)
• Random arrivals, random service times, one
server, finite customer population.
• This model is used if the population is
extraordinarily small.
Arrival Rate of a Customer
• In the M/M/1 with finite population model,
arrival rate  is defined as “arrival rate of a
customer”, or “how often a customer
comes”. For example:
If a customer goes to a barber shop every 15
days, then this customer’s arrival rate is
= 1/15= 0.067 per day = 2 per month.
Example:
Printers Repair p.608
• A printer breaks down randomly. On
average, it breaks down every 20 hours.
• Repair time is random. On average, it takes
2 hours to repair a broken printer.
• Evaluate this printer-service system. (Who
is “customer”?)
• Calculate the total cost if printer downtime
cost is $120/hour, and the technician is paid
$25/hour.