Introduction to Queuing and Simulation. Chapter 6 Business Process Modeling, Simulation and Design. Overview (I). What is queuing/ queuing theory? Why is it an important tool? Examples of different queuing systems Components of a queuing system The exponential distribution & queuing
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design and decision making
Why is Queuing Analysis Important?
Prototype Example – ER at County Hospital
Examples of Real World Queuing Systems?
Components of a Basic Queuing Process
Input Source
The Queuing System
Served Jobs
Service Mechanism
Calling Population
Jobs
Queue
leave the system
Queue Discipline
Arrival Process
Service Process
Queue Configuration
Components of a Basic Queuing Process (II)
Components of a Basic Queuing Process (III)
Ex. Supermarket express lanes
Labor specialization possible
Customer has more flexibility
Balking behavior may be deterred
Several medium-length lines are less intimidating than one very long line
Guarantees fairness
FIFO applied to all arrivals
No customer anxiety regarding choice of queue
Avoids “cutting in” problems
The most efficient set up for minimizing time in the queue
Jockeying (line switching) is avoided
Multiple v.s. Single Customer Queue Configuration
Multiple Line Advantages
Single Line Advantages
Components of a Basic Queuing Process (IV)
Mitigating Effects of Long Queues
A Commonly Seen Queuing Model (I)
The Queuing System
The Service Facility
C S = Server
C S
•
•
•
C S
The Queue
Customers (C)
C C C … C
Customer =C
A Commonly Seen Queuing Model (II)
The Exponential Distribution and Queuing
Definition: A stochastic (or random) variable Texp( ), i.e., is exponentially distributed with parameter , if its frequency function is:
The Cumulative Distribution Function is:
The mean = E[T] = 1/
The Variance = Var[T] = 1/ 2
Properties of the Exp-distribution (I)
P(0Tt) > P(t T t+t) for all t, t0
Properties of the Exp-distribution (II)
Properties of the Exp-distribution (III)
Assume that {T1, T2, …, Tn} represent n independent and exponentially distributed stochastic variables with parameters {1, 2, …, n}.
Properties of the Exp-distribution (IV)
Let X(t) be the number of events occurring in the interval [0,t]. If the time between consecutive events is T and Texp()
t
Stochastic Processes in Continuous Time
Two equivalent definitions of the Poisson Process
b) For a small time interval h it holds that
Properties of the Poisson Process
a) Aggregation of N Poisson processes with intensities
{1, 2, …, n} renders a new Poisson process with intensity = 1+ 2+…+ n.
N(t)= Number of customers/jobs in the system at time t
Pn(t)= The probability that at time t, there are n customers/jobs in the system.
n= Average arrival intensity (= # arrivals per time unit) at n customers/jobs in the system
n= Average service intensity for the system when there are n customers/jobs in it. (Note, the total service intensity for all occupied servers)
= The utilization factor for the service facility. (= The expected fraction of the time that the service facility is being used)
Example – Service Utilization Factor
Queuing Theory Focus on Steady State
With few exceptions Queuing Theory has focused on analyzing steady state behavior
Transient and Steady State Conditions
Notation For Steady State Analysis
Pn = The probability that there are exactly n customers/jobs in the system (in steady state, i.e., when t)
L = Expected number of customers in the system (in steady state)
Lq =Expected number of customers in the queue (in steady state)
W = Expected time a job spends in the system
Wq= Expected time a job spends in the queue
Assumptions
n-1
n
n
0
1
2
1
2
n
n+1
A Birth-and-Death Process Rate Diagram
0
1
n-1
n
= State n, i.e., the case of n customers/jobs in the system
The Rate In = Rate Out Principle:
Mean entrance rate = Mean departure rate
Steady State Analysis of B-D Processes (I)
Steady State Analysis of B-D Processes (III)
Assumptions - the Basic Queuing Process
n+1
1
0
n
2
n-1
Pn = n(1- )
P(nk) = k
P0 = 1-
L=/(1- )
Lq= 2/(1- ) = L-
W=L/=1/(- )
Wq=Lq/= /( (- ))
The M/M/1 Model
Generalization of the M/M/1 model
c+1
c
2
1
0
c-1
c-2
c
c
(c-2)
2
(c-1)
The M/M/c Model (I)
Steady State
Condition:
=(/c)<1
Little’s Formula Wq=Lq/
W=Wq+(1/)
Little’s Formula L=W= (Wq+1/ ) = Lq+ /
Example – ER at County Hospital
Summary of Results – County Hospital
The M/M/c/K model always has a steady state solution since the queue can never “explode”
The M/M/c/K – Model (II)
2
1
0
c-1
K
c
K-1
c
c
3
2
(c-1)
c
The M/M/c/K – Model (III)
1
2
c-1
N
(N-1)
(N-(c-1))
N
c
N-1
c
3
2
(c-1)
c
The M/M/c//N – Model (II)
Priority-Discipline Queuing Models
Queuing Modeling and System Design (I)
Queuing Modeling and System Design (II)
Different Shortage Cost Situations
• Non-profit organizations
Analyzing Design-Cost Tradeoffs
TC
Cost
Min TC = WC + SC
SC
WC
Process capacity
Analyzing Linear Waiting Costs
SC = c*CS()
A Decision Model for System Design
From a structural point of view, a few fast servers are usually better than several slow ones with the same maximum capacity
Determining and c
Example – “Computer Procurement”
are exponential M/M/1 model
Simulation v.s. Symbolic & Analytical Tools
Modern Simulation Software Packages are Breaking Compromises
Steps in a BPD Simulation Project
Phase 1
Problem Definition
1. Problem formulation
2. Set objectives and overall project plan
Phase 2
Model Building
4. Data Collection
3. Model conceptualization
5. Model Translation
Phase 3
Experimentation
No
6. Verified
Yes
No
No
7. Validated
Yes
Phase 4
Implementation
8. Experimental Design
9. Model runs and analysis
No
Yes
11. Documentation, reporting and
implementation
10. More runs
Model Verification and Validation
Conceptual
validation
Calibration and
Validation
Model
verification
Operational Model
(Computerized representation)
Validation - an Iterative Calibration Process
Example – Simulation of a M/M/1 Queue
Lq = 0.832/(1-0.83) 4.2 Wq = Lq/ 4.2/5 0.83