Queueing Theory-1

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Queueing Theory
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Basic Queueing Process
?

• Arrivals
• Arrival time distribution
• Calling population (infinite or finite)

• Service
• Number of servers(one or more)
• Service time distribution

• Queue
• Capacity(infinite or finite)
• Queueing discipline

Queueing System
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Examples and Applications

• Call centers (help desks, ordering goods)
• Manufacturing
• Banks
• Telecommunication networks
• Internet service
• Intelligence gathering
• Restaurants
• Other examples.

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Labeling Convention (Kendall-Lee)
/ / / / /
Interarrival timedistribution
Number of servers
Queueing discipline
System capacity
Calling population size
Service timedistribution
Notes
FCFS first come, first served LCFS last come,
first served SIRO service in random
order GD general discipline
M Markovian (exponential interarrival times,
Poisson number of arrivals) D Deterministic Ek Er
lang with shape parameter k G General
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Labeling Convention (Kendall-Lee)

• ExamplesM/M/1M/M/5M/G/1M/M/3/LCFSEk/G/2//10
  M/M/1///100

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Terminology and Notation

• State of the system Number of customers in the
  queueing system (includes customers in service)
• Queue length Number of customers waiting for
  service
• State of the system - number of customers
  being served
• N(t) State of the system at time t, t 0
• Pn(t) Probability that exactly n customers are
  in the queueing system at time t

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Terminology and Notation

• ?n Mean arrival rate (expected arrivals per
  unit time) of new customers when n customers
  are in the system
• s Number of servers (parallel service
  channels)
• ?n Mean service rate for overall system
  (expected customers completing service per
  unit time) when n customers are in the system
• Note ?n represents the combined rate at which
  all busy servers (those serving customers)
  achieve service completion.

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Terminology and Notation

• When arrival and service rates are constant for
  all n,
• ? mean arrival rate (expected arrivals
  per unit time)
• ? mean service rate for a busy server
• 1/? expected interarrival time
• 1/? expected service time
• ? ?/s? utilization factor for the service
  facility expected fraction of time the
  systems service capacity (s?) is being utilized
  by arriving customers (?)

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Terminology and NotationSteady State

• When the system is in steady state, then
• Pn probability that exactly n customers are
  in the queueing system
• L expected number of customers in queueing
  system
• Lq expected queue length (excludes customers
  being served)

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Terminology and NotationSteady State

• When the system is in steady state, then
• ? waiting time in system (includes service
  time) for each individual customer
• W E?
• ?q waiting time in queue (excludes service
  time) for each individual customer
• Wq E?q

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Littles Formula
Demonstrates the relationships between L, W, Lq,
and Wq

• Assume ?n? and ?n? (arrival and service rates
  constant for all n)
• In a steady-state queue,

Intuitive Explanation
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Littles Formula (continued)

• This relationship also holds true for ? (expected
  arrival rate) when ?n are not equal.

Recall, Pn is the steady state probability of
having n customers in the system
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Heading toward M/M/s

• The most widely studied queueing models are of
  the form M/M/s (s1,2,)
• What kind of arrival and service distributions
  does this model assume?
• Reviewing the exponential distribution.
• If T exponential(a), then
• A picture of the distribution

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Exponential Distribution Reviewed

• If T exponential(?), then

Var(T) ______
ET ______
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Property 1Strictly Decreasing

• The pdf of exponential, fT(t), is a strictly
  decreasing function
• A picture of the pdf

fT(t)
?
t
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Property 2Memoryless

• The exponential distribution has lack of memory
• i.e. P(T gt ts T gt s) P(T gt t) for all
  s, t 0.
• Example
• P(T gt 15 min T gt 5 min) P(T gt 10 min)
• The probability distribution has no memory of
  what has alreadyoccurred.

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Property 2Memoryless

• Prove the memoryless property
• Is this assumption reasonable?
• For interarrival times
• For service times

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Property 3Minimum of Exponentials

• The minimum of several independent exponential
  random variables has an exponential distribution
• If T1, T2, , Tn are independent r.v.s, Ti
  expon(?i) and
• U min(T1, T2, , Tn),
• U expon( )
• Example
• If there are n servers, each with exponential
  service times with mean ?, then U time until
  next service completion expon(____)

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Property 4Poisson and Exponential

• If the time between events, Xn expon(?),
  thenthe number of events occurring by time t,
  N(t) Poisson(?t)
• Note
• EX(t) at, thus the expected number of events
  per unit time is a

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Property 5Proportionality

• For all positive values of t, and for small ?t,
• P(T t?t T gt t) ??t
• i.e. the probability of an event in interval ?t
  is proportional to the length of that interval

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Property 6Aggregation and Disaggregation

• The process is unaffected by aggregation and
  disaggregation

Aggregation
Disaggregation
N1 Poisson(?1)
N1 Poisson(?p1)
p1
N2 Poisson(?2)
N2 Poisson(?p2)
N Poisson(?)
N Poisson(?)
p2


pk
? ?1?2?k
Nk Poisson(?k)
Nk Poisson(?pk)
Note p1p2pk1
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Back to Queueing

• Remember that N(t), t 0, describes the state of
  the systemThe number of customers in the
  queueing system at time t
• We wish to analyze the distribution of N(t) in
  steady state

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Birth-and-Death Processes

• If the queueing system is M/M////, N(t) is a
  birth-and-death process
• A birth-and-death process either increases by 1
  (birth), or decreases by 1 (death)
• General assumptions of birth-and-death processes
• 1. Given N(t) n, the probability distribution
  of the time remaining until the next birth is
  exponential with parameter ?n
• 2. Given N(t) n, the probability distribution
  of the time remaining until the next death is
  exponential with parameter µn
• 3. Only one birth or death can occur at a time

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Rate Diagrams
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Steady-State Balance Equations
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M/M/1 Queueing System

• Simplest queueing system based on birth-and-death
• We define ? mean arrival rate? mean service
  rate ? ? / ? utilization ratio
• We require ? lt ? , that is ? lt 1 in order to have
  a steady state
• Why?

Rate Diagram
?
?
?
?
?
1
2
3
4

0
?
?
?
?
?
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M/M/1 Queueing System Steady-State Probabilities

• Calculate Pn, n 0, 1, 2,

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M/M/1 Queueing System L, Lq, W, Wq

• Calculate L, Lq, W, Wq

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M/M/1 Example ER

• Emergency cases arrive independently at random
• Assume arrivals follow a Poisson input process
  (exponential interarrival times) and that the
  time spent with the ER doctor is exponentially
  distributed
• Average arrival rate 1 patient every ½ hour
• ?
• Average service time 20 minutes to treat each
  patient
• ?
• Utilization
• ?

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M/M/1 Example ERQuestions

• What is the
• probability that the doctor is idle?
• probability that there are n patients?
• expected number of patients in the ER?
• expected number of patients waiting for the
  doctor?
• expected time in the ER?
• expected waiting time?
• probability that there are at least two patients
  waiting?
• probability that a patient waits more than 30
  minutes?

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Car Wash Example

• Consider the following 3 car washes
• Suppose cars arrive according to a Poisson input
  process and service follows an exponential
  distribution
• Fill in the following table

? ? L Lq W Wq P0
Car Wash A 0.1 car/min 0.5 car/min
Car Wash B 0.1 car/min 0.11 car/min
Car Wash C 0.1 car/min 0.1 car/min
What conclusions can you draw from your results?
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M/M/s Queueing System

• We define ? mean arrival rate? mean service
  rate s number of servers (s gt 1)? ? / s?
  utilization ratio
• We require ? lt s? , that is ? lt 1 in order to
  have a steady state

Rate Diagram
1
2
3
4

0
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M/M/s Queueing System Steady-State Probabilities
Pn CnP0
and
where
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M/M/s Queueing System L, Lq, W, Wq
How to find L? W? Wq?
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M/M/s Example A Better ER

• As before, we have
• Average arrival rate 1 patient every ½ hour?
  2 patients per hour
• Average service time 20 minutes to treat each
  patient? 3 patients per hour
• Now we have 2 doctorss
• Utilization
• ?

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M/M/s Example ERQuestions

• What is the
• probability that both doctors are idle?
• probability that exactly one doctor is idle?
• probability that there are n patients?
• expected number of patients in the ER?
• expected number of patients waiting for a doctor?
• expected time in the ER?
• expected waiting time?
• probability that there are at least two patients
  waiting?
• probability that a patient waits more than 30
  minutes?

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Performance Measurements s 1 s 2
? 2/3 1/3
L 2 3/4
Lq 4/3 1/12
W 1 hr 3/8 hr
Wq 2/3 hr 1/24 hr
P(at least two patients waiting in queue) 0.296 0.0185
P(a patient waits more than 30 minutes) 0.404 0.022
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Travel Agency Example

• Suppose customers arrive at a travel agency
  according to a Poisson input process and service
  times have an exponential distribution
• We are given
• ? .10/minute 1 customer every 10 minutes
• ? .08/minute 8 customers every 100 minutes
• If there were only one server, what would happen?
• How many servers would you recommend?

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Single Queue vs. Multiple Queues

• Would you ever want to keep separate queues for
  separate servers?

Single queue
Multiple queues
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Bank Example

• Suppose we have two tellers at a bank
• Compare the single server and multiple server
  models
• Assume ? 2, ? 3

L Lq W Wq P0


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Bank ExampleContinued

• Suppose we now have 3 tellers
• Again, compare the two models

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M/M/s//K Queueing Model(Finite Queue Variation
of M/M/s)

• Now suppose the system has a maximum capacity, K
• We will still consider s servers
• Assuming s K, the maximum queue capacity is K
  s
• List some applications for this model
• Draw the rate diagram for this problem

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M/M/s//K Queueing Model(Finite Queue Variation
of M/M/s)
Rate Diagram
1
2
3
4

0

• Balance equations Rate In Rate Out

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M/M/s//K Queueing Model(Finite Queue Variation
of M/M/s)

• Solving the balance equations, we get the
  following steady state probabilities

Verify that these equations match those given in
the text for the single server case (M/M/1//K)
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M/M/s//K Queueing Model(Finite Queue Variation
of M/M/s)
To find W and Wq Although L ? lW and Lq ? lWq
because ln is not equal for all n,
and
where
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M/M/s///N Queueing Model(Finite Calling
Population Variation of M/M/s)

• Now suppose the calling population is finite
• We will still consider s servers
• Assuming s K, the maximum queue capacity is K
  s
• List some applications for this model
• Draw the rate diagram for this problem

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M/M/s///N Queueing Model(Finite Calling
Population Variation of M/M/s)
Rate Diagram
1
2
3
4

0

• Balance equations Rate In Rate Out

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M/M/s///N Results
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Queueing Models with Nonexponential Distributions

• M/G/1 Model
• Poisson input process, general service time
  distribution with mean 1/? and variance ?2
• Assume ? ?/? lt 1
• Results

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Queueing Models with Nonexponential Distributions

• M/Ek/1 Model
• Erlang Sum of exponentials
• Think it would be useful?
• Can readily apply the formulae on previous slide
  where
• Other models
• M/D/1
• Ek/M/1
• etc

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Application of Queueing Theory

• We can use the results for the queueing models
  when making decisions on design and/or operations
• Some decisions that we can address
• Number of servers
• Efficiency of the servers
• Number of queues
• Amount of waiting space in the queue
• Queueing disciplines

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Number of Servers

• Suppose we want to find the number of servers
  that minimizes the expected total cost, ETC
• Expected Total Cost Expected Service Cost
  Expected Waiting Cost(ETC ESC EWC)
• How do these costs change as the number of
  servers change?

Expected cost
Number of servers
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Repair Person Example

• SimInc has 10 machines that break down frequently
  and 8 operators
• The time between breakdowns Exponential, mean
  20 days
• The time to repair a machine Exponential, mean
  2 days
• Currently SimInc employs 1 repair person and is
  considering hiring a second
• Costs
• Each repair person costs 280/day
• Lost profit due to less than 8 operating
  machines400/day for each machine that is down
• Objective Minimize total cost
• Should SimInc hire the additional repair person?

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Repair Person ExampleProblem Parameters

• What type of problem is this?
• M/M/1
• M/M/s
• M/M/s/K
• M/G/1
• M/M/s finite calling population
• M/Ek/1
• M/D/1
• What are the values of ? and ??

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Repair Person ExampleRate Diagrams

• Draw the rate diagram for the single-server and
  two-server case

Single server
1
2
3
4
10
9
8
0

1
2
3
4
10
9
8
Two servers
0

• Expected service cost (per day) ESC
• Expected waiting cost (per day) EWC

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Repair Person ExampleSteady-State Probabilities

• Write the balance equations for each case
• How to find EWC for s1? s2?

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Repair Person ExampleEWC Calculations
Nn g(n) s1 s1 s2 s2
Nn g(n) Pn g(n) Pn Pn g(n) Pn
0 0 0.271 0 0.433 0
1 0 0.217 0 0.346 0
2 0 0.173 0 0.139 0
3 400 0.139 56 0.055 24
4 800 0.097 78 0.019 16
5 1200 0.058 70 0.006 8
6 1600 0.029 46 0.001 0
7 2000 0.012 24 0.0003 0
8 2400 0.003 7 0.00004 0
9 2800 0.0007 0 0.000004 0
10 3200 0.00007 0 0.0000002 0
EWC EWC 281/day 281/day 48/day 48/day
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Repair Person ExampleResults

• We get the following results

s ESC EWC ETC
1 280/day 281/day 561/day
2 560/day 48/day 608/day
3 840/day 0/day 840/day

• What should SimInc do?

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Supercomputer Example

• Emerald University has plans to lease a
  supercomputer
• They have two options

Supercomputer Mean number of jobs per day Cost per day
MBI 30 jobs/day 5,000/day
CRAB 25 jobs/day 3,750/day

• Students and faculty jobs are submitted on
  average of 20 jobs/day, distributed
  Poisson i.e. Time between submissions
  __________
• Which computer should Emerald University lease?

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Supercomputer ExampleWaiting Cost Function

• Assume the waiting cost is not linear h(?)
  500 ? 400 ? 2 (? waiting time in days)
• What distribution do the waiting times follow?
• What is the expected waiting cost, EWC?

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Supercomputer ExampleResults

• Next incorporate the leasing cost to determine
  the expected total cost, ETC
• Which computer should the university lease?