SBEN419 – Operations Research
in Healthcare
Lecture 11
Introduction to Queuing Models
Today’s lecture
• Introduction to queuing theory
• The Exponential and Poisson distribution
functions
• The Birth-Death Process
Queuing theory
• Queueing theory is the mathematical study of
waiting lines, or queues.
• Originated in the early 20th century by A. K.
Erlang who created models to describe the
Copenhagen telephone switchboards are
operated for Bell labs (USA).
(https://www.youtube.com/watch?v=xJ1fKFqt7qU&t=179s)
• Queuing models are based on stochastic
processes that can be described using probability
distribution functions that represent different
random variables related to the studied system.
Queuing system as a stochastic
process
Arrival
process
Departure
process
Waiting line (queue)
Arrival process: time between arrivals (interarrival time) is a random
variable.
Departure process: service time is a random variable.
The queuing system can be viewed as a stochastic process.
What is a stochastic process?
Stochastic process: System that changes over
time in an uncertain manner
State: Snapshot of the system at some fixed
point in time
Transition: Movement from one state to
another
Components of Stochastic Model
Time: Either continuous or discrete parameter.
t0 t1
t2
t3 t4
time
State: Describes the attributes of a system at some point in time.
s = (s1, s2, . . . , sv); for ATM example s = (n)
Where n is the number of people in the system.
• Convenient to assign a unique nonnegative integer index to
each possible value of the state vector. We call this X and
require that for each s → X.
• For ATM example, X = n.
• In general, Xt is a random variable.
The ATM example of lecture 2
Clock
Q(t)
N(t)
0
0
1
1.73
1
2
2.9
0
1
3.08 3.79 4.41 4.66 8.05 12.57 17.03 18.69 19.39 23.05 25.12
1
2
3
2
1
0
0
0
1
0
0
2
3
4
3
2
1
0
1
2
1
0
Since time intervals from one state to the next are not equal in length, we have a
continuous-time stochastic process.
Exponential Distribution…
• Exponential distribution has a probability density function
that is given as:
• Note that x ≥ 0. Time (for example) is a non-negative quantity; the
exponential distribution is often used for time related phenomena such as
the length of time between phone calls or between parts arriving at an
assembly station. Note also that the mean and standard deviation are
equal to each other and to the inverse of the parameter of the distribution
(lambda )
E(x) = SD(x)
Exponential Distribution…
The exponential distribution depends upon the value of
Smaller values of “flatten” the curve:
(E.g. exponential
distributions for
= (.5, 1, 2)
Lack-of-memory property
P T  t + s ¨T  t =
=
e
− t
P t  T  t + s
−  (t + s )
−e
e− t
P T  t
= 1− e
−s
= P T  s
Poisson Probability Distribution
• Poisson Probability Function
where:
f(x) = probability of x occurrences in an interval
 = mean number of occurrences in an interval
e = Euler’s number which is approximately
= 2.71828…
Poisson Probability Distribution
• Expected Value
E(x) = m = 
• Variance
Var(x) =  2 = 
• Standard Deviation
Poisson Probability Distribution
• Properties of a Poisson Experiment
– The probability of an occurrence is the same for
any two intervals of equal length.
– The occurrence or nonoccurrence in any interval is
independent of the occurrence or nonoccurrence
in any other interval.
Chapter 3
Statistical Quality Control, 7th Edition by Douglas C. Montgomery.
Copyright (c) 2013 John Wiley & Sons, Inc.
In-class assignment
The birth-death process
• Most elementary queueing models assume
that the inputs (arriving customers) and
outputs (leaving customers) of the queueing
system occur according to the birth-and-death
process.
• the term birth refers to the arrival of a new
customer into the queueing system, and
death refers to the departure of a served
customer.
Elements of the birth-death process
• The state of the system at time t (t ≥ 0),
denoted by N(t), is the number of customers
in the queueing system at time t.
• individual births and deaths occur randomly,
where their mean occurrence rates depend
only upon the current state of the system
(Markovian property).
Assumptions of Birth-Death Process
1. Given N(t) = n, the current probability distribution of
the remaining time until the next birth (arrival) is
exponential with parameter n (n = 0, 1, 2, . . .).
2. Given N(t) = n, the current probability distribution of
the remaining time until the next death (service
completion) is exponential with parameter mn (n = 1,
2, . . .).
3. Both random variables are mutually independent.
Steady-state properties of birth-death process
Steady-state properties of birth-death process
Steady-state properties of birth-death process
Steady-state properties of birth-death process
Steady-state properties
Steady-state performance measures
for queuing systems
Relationships between performance measures
Performance measures for the birthdeath process