COMM 225 — Production and Operations Management
CHAPTER 18: Waiting-Line Analysis
Comprehensive Exam Preparation Notes
1. Introduction
Waiting lines occur when there is a temporary imbalance between supply (capacity) and
demand. They add to the cost of operation and reflect negatively on customer service. From a
managerial perspective, the key is to find the balance that provides an adequate level of
service at a reasonable cost. Waiting lines occur not only with people but with orders waiting to
be filled, trucks waiting to be unloaded, jobs waiting to be processed, equipment awaiting
repair, and planes waiting to land.
Queueing theory is a mathematical approach to the analysis of waiting lines, applicable to call
centres, banks, post offices, restaurants, theme parks, telecommunications, and traffic
management. Modern queueing theory is based on studies about automatic dialing equipment
made by Danish telephone engineer A. K. Erlang.
KEY Queueing Theory: A mathematical approach to the analysis of waiting lines
TERM (queues). Used to balance the cost of providing service capacity against the cost
of having customers wait.
Why Is There Waiting?
Many people are surprised that waiting lines form even though a system is underloaded. For
example, a fast-food restaurant with capacity for 200 orders/hour may still experience waiting
lines when demand is only 150 orders/hour. The reason is that both arrival times and service
durations exhibit high variability. Customers arrive at random rather than at evenly spaced
intervals, and some orders take longer than others. As a result, the system at times becomes
temporarily overloaded (from a micro standpoint), generating waiting lines, while at other
times is idle. In systems where variability is minimal or non-existent (arrivals can be scheduled,
service durations are constant), waiting lines do not ordinarily form.
Goal of Waiting-Line Analysis
The goal of queueing analysis is to minimize total costs. There are two basic categories of
costs:
Cost Category Definition Examples
Costs of Costs incurred due Salaries paid to employees while waiting
Customer Waiting to customers (or (mechanics waiting for tools), cost of waiting
jobs) waiting for space (size of doctor's waiting room, driveway at
service. car wash), fuel consumed by planes waiting to
land, loss of business due to customers refusing
to wait and going elsewhere.
Capacity Costs Costs of Number of bays at a car wash, number of
, Cost Category Definition Examples
maintaining the checkouts at a supermarket, number of repair
ability to provide people, number of highway lanes. When a service
service. facility is idle, capacity is lost since it cannot be
stored.
As capacity increases, capacity cost increases (approximately linearly), while customer waiting
cost decreases (non-linearly, at a faster rate initially). Total cost is a U-shaped curve and the
goal is to identify the level of service capacity that minimizes total cost. Unlike the EOQ model,
the minimum point on the total cost curve is NOT necessarily where the two cost lines
intersect.
Psychology of Waiting
Despite best efforts, in some instances it is not feasible to shorten waiting times. However,
steps can be taken to make waiting more acceptable by reducing perceived waiting time
relative to actual waiting time. Managerial suggestions:
• Determine the acceptable wait time for the operation. Try to keep waiting time
experienced by a customer consistent over time.
• Install distractions that entertain and involve customers (magazines, TV, mirrors outside
elevators; ask the customer to fill out a form and use the lounge before eating).
• Inform the customer of the cause of an abnormal wait and peak times. Keep the line
moving continuously.
• Use first-come, first-served discipline (fairness is important to customers).
• Allow customers to serve themselves. Prepare customers for service before the actual
service (e.g., encourage customers to print boarding pass at home the night before a
flight).
• Make people conscious of time only if they overestimate the wait time. Keep staff who
are not serving customers out of sight.
• Segment customers by personality: 'watchers' actually like to spend time in lines;
'impatients' need shorter lines. Airlines and hotels use club memberships for express
check-in.
• Disney's approach: Provide distractions (characters, videos, vendors), alternatives to
waiting in line (FastPass — reserve time later in the day for popular rides), keep
customers informed (signs with approximate wait times), exceed expectations (actual
wait shorter than posted times), maintain comfortable waiting environments (often
inside, protected from weather). Disney presents seminars to other companies on
managing waiting lines.
• Six Flags Flash Pass: Allows customers to reserve their place in line without physically
waiting, freeing them to do other activities. Alert notifies when their turn comes.
Constraint Management
Managers may be able to reduce waiting times by actively managing constraints. In the short
term, facility size is fixed. Options include:
, • Use temporary workers: Temporary or part-time workers during busy periods. Trade-
offs: training costs, quality issues, possibly slower service.
• Shift demand: Variable pricing strategies smooth demand more evenly. Theatres lower
prices to shift demand from busy to slow times. Restaurants offer 'early-bird specials.'
Retail businesses offer coupons valid only for certain slow days or times.
• Standardize the service: The more the service can be standardized, the shorter the
waiting line.
• Look for a bottleneck: One part of a process may be largely responsible for a slow
service rate; improving that part might yield a disproportionate increase in the service
rate. Employees often have insights that can be exploited.
2. Queueing System Characteristics and Performance Measures
Model choice is dependent on the characteristics of the waiting-line system. The four main
characteristics are: (1) potential number of customers, (2) number of servers and structure of
queueing system, (3) arrival and service patterns, and (4) queue discipline.
Potential Number of Customers
Source Type Description Examples
Infinite Source The potential number of Supermarkets, drugstores, banks,
customers greatly exceeds restaurants, theatres, amusement parks,
system capacity. call centres.
Theoretically, a large number
of customers can request
service at any given time.
Finite Source The potential number of Repair technician responsible for a certain
customers is limited. The number of machines; operator loading
arrival rate of customers is and unloading a bank of 4 machines;
affected by the length of the nurse answering patient calls for a 10-bed
waiting line (arrival rate ward; company truck maintenance shop
decreases as line length for 20 trucks.
increases because fewer units
are 'out and running').
Number of Servers and Structure of Queueing System
Each server can handle one customer at a time. A group of servers working together as a
team (e.g., surgical team) is treated as a single server. Systems can have single or multiple
servers and single or multiple phases (stages). This chapter focuses on single-phase
systems.
Configuration Description Examples
Single Server, Single One server, one stage Small grocery with one cashier,
Phase single-bay car wash
CHAPTER 18: Waiting-Line Analysis
Comprehensive Exam Preparation Notes
1. Introduction
Waiting lines occur when there is a temporary imbalance between supply (capacity) and
demand. They add to the cost of operation and reflect negatively on customer service. From a
managerial perspective, the key is to find the balance that provides an adequate level of
service at a reasonable cost. Waiting lines occur not only with people but with orders waiting to
be filled, trucks waiting to be unloaded, jobs waiting to be processed, equipment awaiting
repair, and planes waiting to land.
Queueing theory is a mathematical approach to the analysis of waiting lines, applicable to call
centres, banks, post offices, restaurants, theme parks, telecommunications, and traffic
management. Modern queueing theory is based on studies about automatic dialing equipment
made by Danish telephone engineer A. K. Erlang.
KEY Queueing Theory: A mathematical approach to the analysis of waiting lines
TERM (queues). Used to balance the cost of providing service capacity against the cost
of having customers wait.
Why Is There Waiting?
Many people are surprised that waiting lines form even though a system is underloaded. For
example, a fast-food restaurant with capacity for 200 orders/hour may still experience waiting
lines when demand is only 150 orders/hour. The reason is that both arrival times and service
durations exhibit high variability. Customers arrive at random rather than at evenly spaced
intervals, and some orders take longer than others. As a result, the system at times becomes
temporarily overloaded (from a micro standpoint), generating waiting lines, while at other
times is idle. In systems where variability is minimal or non-existent (arrivals can be scheduled,
service durations are constant), waiting lines do not ordinarily form.
Goal of Waiting-Line Analysis
The goal of queueing analysis is to minimize total costs. There are two basic categories of
costs:
Cost Category Definition Examples
Costs of Costs incurred due Salaries paid to employees while waiting
Customer Waiting to customers (or (mechanics waiting for tools), cost of waiting
jobs) waiting for space (size of doctor's waiting room, driveway at
service. car wash), fuel consumed by planes waiting to
land, loss of business due to customers refusing
to wait and going elsewhere.
Capacity Costs Costs of Number of bays at a car wash, number of
, Cost Category Definition Examples
maintaining the checkouts at a supermarket, number of repair
ability to provide people, number of highway lanes. When a service
service. facility is idle, capacity is lost since it cannot be
stored.
As capacity increases, capacity cost increases (approximately linearly), while customer waiting
cost decreases (non-linearly, at a faster rate initially). Total cost is a U-shaped curve and the
goal is to identify the level of service capacity that minimizes total cost. Unlike the EOQ model,
the minimum point on the total cost curve is NOT necessarily where the two cost lines
intersect.
Psychology of Waiting
Despite best efforts, in some instances it is not feasible to shorten waiting times. However,
steps can be taken to make waiting more acceptable by reducing perceived waiting time
relative to actual waiting time. Managerial suggestions:
• Determine the acceptable wait time for the operation. Try to keep waiting time
experienced by a customer consistent over time.
• Install distractions that entertain and involve customers (magazines, TV, mirrors outside
elevators; ask the customer to fill out a form and use the lounge before eating).
• Inform the customer of the cause of an abnormal wait and peak times. Keep the line
moving continuously.
• Use first-come, first-served discipline (fairness is important to customers).
• Allow customers to serve themselves. Prepare customers for service before the actual
service (e.g., encourage customers to print boarding pass at home the night before a
flight).
• Make people conscious of time only if they overestimate the wait time. Keep staff who
are not serving customers out of sight.
• Segment customers by personality: 'watchers' actually like to spend time in lines;
'impatients' need shorter lines. Airlines and hotels use club memberships for express
check-in.
• Disney's approach: Provide distractions (characters, videos, vendors), alternatives to
waiting in line (FastPass — reserve time later in the day for popular rides), keep
customers informed (signs with approximate wait times), exceed expectations (actual
wait shorter than posted times), maintain comfortable waiting environments (often
inside, protected from weather). Disney presents seminars to other companies on
managing waiting lines.
• Six Flags Flash Pass: Allows customers to reserve their place in line without physically
waiting, freeing them to do other activities. Alert notifies when their turn comes.
Constraint Management
Managers may be able to reduce waiting times by actively managing constraints. In the short
term, facility size is fixed. Options include:
, • Use temporary workers: Temporary or part-time workers during busy periods. Trade-
offs: training costs, quality issues, possibly slower service.
• Shift demand: Variable pricing strategies smooth demand more evenly. Theatres lower
prices to shift demand from busy to slow times. Restaurants offer 'early-bird specials.'
Retail businesses offer coupons valid only for certain slow days or times.
• Standardize the service: The more the service can be standardized, the shorter the
waiting line.
• Look for a bottleneck: One part of a process may be largely responsible for a slow
service rate; improving that part might yield a disproportionate increase in the service
rate. Employees often have insights that can be exploited.
2. Queueing System Characteristics and Performance Measures
Model choice is dependent on the characteristics of the waiting-line system. The four main
characteristics are: (1) potential number of customers, (2) number of servers and structure of
queueing system, (3) arrival and service patterns, and (4) queue discipline.
Potential Number of Customers
Source Type Description Examples
Infinite Source The potential number of Supermarkets, drugstores, banks,
customers greatly exceeds restaurants, theatres, amusement parks,
system capacity. call centres.
Theoretically, a large number
of customers can request
service at any given time.
Finite Source The potential number of Repair technician responsible for a certain
customers is limited. The number of machines; operator loading
arrival rate of customers is and unloading a bank of 4 machines;
affected by the length of the nurse answering patient calls for a 10-bed
waiting line (arrival rate ward; company truck maintenance shop
decreases as line length for 20 trucks.
increases because fewer units
are 'out and running').
Number of Servers and Structure of Queueing System
Each server can handle one customer at a time. A group of servers working together as a
team (e.g., surgical team) is treated as a single server. Systems can have single or multiple
servers and single or multiple phases (stages). This chapter focuses on single-phase
systems.
Configuration Description Examples
Single Server, Single One server, one stage Small grocery with one cashier,
Phase single-bay car wash
