Queueing Theory: Part I

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Queueing Theory Part I
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The Basic Structure of Queueing Models
The Basic Queueing Process Customers are
generated over time by an input source. The
customers enter a queueing system. A required
service is performed in the service mechanism.
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Input source
Queueing system
Served customers
Service mechanism
Queue
Customers
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Input Source (Calling Population) The size of
Input Source (Calling Population) is assumed
infinite because the calculations are far
easier. The pattern by which customers are
generated is assumed to be a Poisson process.
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The probability distribution of the time between
consecutive arrivals is an exponential
distribution. The time between consecutive
arrivals is referred to as the interarrival time.
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Queue The queue is where customers wait before
being served. A queue is characterized by the
maximum permissible number of customers that it
can contain. The assumption of an infinite queue
is the standard one for most queueing models.
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Queue Discipline The queue discipline refers to
the order in which members of the queue are
selected for service. For example, (a)
First-come-first-served (b) Random
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Service Mechanism The service mechanism consists
of one or more service facilities, each of which
contains one or more parallel service channels,
called servers. The time at a service facility is
referred to as the service time. The service-time
is assumed to be the exponential distribution.
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Elementary Queueing Process
Served customers
Queueing system
Queue
C C C C
S S Service S facility S
Customers
C C C C C C C
Served customers
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Distribution of service times
Number of servers
Distribution of interarrival times
Where M exponential distribution
(Markovian) D degenerate distribution
(constant times) Erlang distribution
(shape parameter k) G general
distribution(any arbitrary
distribution allowed)
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Both interarrival and service times have an
exponential distribution. The number of servers
is k . Interarrival time is an exponential
distribution. No restriction on service time. The
number of servers is exactly 1.
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Terminology and Notation State of system of
customers in queuing system. Queue length
of customers waiting for
service to begin. N(t)
of customers in queueing
system at time t (t 0)
probability of exactly n customers
in queueing system at time
t.
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of servers in queueing system. A mean arrival
rate (the expected number of arrivals per unit
time) of new customers when n customers are in
system. A mean service rate for overall system
(the expected number of customers completing
service per unit time) when n customers are in
system. Note represents a combined rate at
which all busy servers (those serving customers)
achieve service completions.
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When is a constant for all n, it is
expressed by When the mean service rate per
busy server is a constant for all n 1, this
constant is denoted by . Under these
circumstances, and are the
expected interarrival time and the expected
service time. is the
utilization factor for the service facility.
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The state of the system will be greatly affected
by the initial state and by the time that has
since elapsed. The system is said to be in a
transient condition. After sufficient time has
elapsed, the state of the system becomes
essentially independent of the initial state and
the elapsed time. The system has reached a
steady-state condition, where the probability
distribution of the state of the system remains
the same over time.
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The probability of exactly n customers in
queueing system. The expected number of
customers in queueing system The expected queue
length (excludes customers being served)
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A waiting time in system (includes service time)
for each individual customer. A waiting time in
queue (excludes service time) for each individual
customer.
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Relationships between and
Assume that is a constant for all
n. In a steady-state queueing process,
Assume that the mean service time is a constant,
for all It follows that,
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The Role of the Exponential Distribution
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An exponential distribution has the following
probability density function
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Relationship to the Poisson distribution
Suppose that the time between consecutive
arrivals has an exponential distribution with
parameter . Let X(t) be the number of
occurrences by time t (t 0) The number
of arrivals follows
(11.1) (11.2)
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The Birth-and-Death Process Most elementary
queueing models assume that the inputs and
outputs of the queueing system occur according to
the birth-and-death process. In the context of
queueing theory, 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.
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The birth-and-death process is a special type of
continuous time Markov chain.
State 0 1 2 3 n-2 n-1
n n1
and are mean rates.
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Rate In Rate Out Principle. For any state of
the system n (n 0,1,2,), average entering
rate average leaving rate. The equation
expressing this principle is called the balance
equation for state n.
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Rate In Rate Out
State 0 1 2 n 1 n
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State
0
1
2
To simplify notation, let
for n 1,2,
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and then define for n 0. Thus,
the steady-state probabilities are
for n 0,1,2,
The requirement that
implies that
so that
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The definitions of L and specify that
is the average arrival rate. is the
mean arrival rate while the system is in state n.
is the proportion of time for state n,
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The M/M/s Model A M/M/s model assumes that all
interarrival times are independently and
identically distributed according to an
exponential distribution, that all service times
are independent and identically distributed
according to another exponential distribution,
and that the number of service is s (any positive
integer).
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In this model the queueing systems mean arrival
rate and mean service rate per busy server are
constant ( and ) regardless of the state
of the system.
(a) Single-server case (s1)


State 0 1 2 3 n-2 n-1
n n1
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(b) Multiple-server case (k gt 1)
for n 0,1,2,
for
n 1,2,k
for n k, k1,...
State 0 1 2 3 k-2 k-1
k k1
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When the maximum mean service rate
exceeds the mean arrival rate, that is, when
a queueing system fitting this model will
eventually reach a steady-state condition.
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When , the mean arrival rate exceeds
the mean service rate, the preceding solution
blows up and grow without bound. Assuming
, we can derive the probability
distribution of the waiting time in the system
(w) for a random arrival when the queue
discipline is first-come-first-served. If this
arrival finds n customers in the system, then the
arrival will have to wait through n 1
exponential service time, including his or her
own.
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1 Single-Server case ( k 1)
Birth-Death Process
0 1 2 n-1 n n1
Rate In Rate Out
State 0 1 2 n
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Similarly,
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Example
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2 Multiple-Server case ( k gt 1)
Birth-Death Process
0 1 2 3 k-2 k-1
k k1
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Rate In Rate Out
State 0 1 2 k - 1 k k 1
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Home Work

• Problem 11-3
• Problem 11-9
• Problem 11-11
• Due date October 28

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Assessment I

• Please indicate the current level of your
  knowledge. (1 no idea, 2 little, 3
  considerable, 4 very well).
• Topic Your Assessment
• (1) Linear Programming
• (2) Dual and Primal Relationship
• (3) Simplex Method
• (4) Data Envelopment Analysis
• (5) PERT/CPM
• (6) Inventory
• Return the assessment by Oct 23 (noon) to
  toshi_at_nmt.edu

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Assessment II

• Please indicate the current level of your
  knowledge. (1 no idea, 2 little, 3
  considerable, 4 very well).
• Topic Your Assessment
• (1) Queuing
• (2) Decision Analysis
• (3) Multi-Criteria Decision Making
• (4) Forecasting
• (5) Markov Process
• Return the assessment by Oct 23 (noon) to
  toshi_at_nmt.edu