Queueing Systems Part II

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Queueing SystemsPart II

• J. M. Akinpelu

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M/M/1/K Queue

• M/M/1/K queueing system
• exponential interarrival distribution
• exponential service distribution
• 1 server
• finite queue (K spaces)
• FCFS service discipline

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M/M/1/K Queue
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M/M/1/K Queue

• It follows that
• Since then, if ? lt ?,
• and

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M/M/1/K Queue

• The utilization of the server is given by
• The expected number in the system is

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M/M/1/K Queue

• We can derive the expected delay using Littles
  Theorem if we interpret the arrival rate
  correctly
• or

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Multiserver Queues

• Now lets assume there are s gt 1 servers
• We will consider two systems
• The Erlang Loss System
• The Erlang Delay System

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The Erlang Loss System

• Blocked Calls Cleared The Erlang Loss System
• M/M/s/s
• exponential interarrival time
• exponential service time
• s servers
• no queueing
• Customers enter the system if at least one of the
  servers is free

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The Erlang Loss System
?j-1?
?j?
j1
j
j?1
?j1 ( j1)?
?j j?
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The Erlang Loss System

• It follows that
• and

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The Erlang Loss System

• The probability that an arriving customer is lost
  is given by
• This is called the Erlang-B formula (named after
  the Danish mathematician A. K. Erlang).

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The Erlang Loss System

• The Erlang loss system is used in engineering
  telephone networks to provide some grade of
  service. The telephone lines are the servers.
• Let
• ? the rate of calls between two exchanges
• 1/? the mean length of a telephone call
• pe the engineered blocking probability

s telephone lines
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The Erlang Loss System

• We want to determine the number of telephone
  lines needed to meet the engineered blocking
  probability pe.
• To do so, we find the smallest number of lines s
  satisfying

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The Erlang Delay System

• Blocked Calls Delayed The Erlang Delay System
• M/M/s/?
• exponential interarrival time
• exponential service time
• s servers
• infinite queue
• Customers enter the queue if all servers are busy

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The Erlang Delay System
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The Erlang Delay System

• Letting we have

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The Erlang Delay System

• The probability that an arriving customer has to
  join the queue is given by
• This is called the Erlang-C formula.

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The Erlang Delay System

• Consider a call center where customers queue
  until there is an available agent.
• Let
• ? the rate of arriving customers
• 1/? the mean time required to serve a customer
• pq the engineered queueing probability

s agents
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The Erlang Delay System

• We want to determine the number of agents needed
  to meet the engineered queueing probability pq.
• To do so, find the smallest number of agents s
  satisfying

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A. K. Erlang

• Agner Krarup Erlang (January 1, 1878 - February
  3, 1929) was a Danish mathematician,
  statistician, and engineer who invented the
  fields of queueing theory and traffic
  engineering.
• Erlang was the first person to study the problem
  of telephone networks. By studying a village
  telephone exchange he worked out a formula, now
  known as Erlang's formula, to calculate the
  fraction of callers attempting to call someone
  outside the village that must wait because all of
  the lines are in use.
• The British Post Office accepted his formula as
  the basis for calculating circuit facilities. The
  mathematics underlying today's complex telephone
  networks is still based on his work. The unit
  of communication activity in these fields is now
  known as the erlang, in recognition of his
  achievements. His name is also given to the
  statistical probability distribution that arises
  from his work. 

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Comparing Single Server and Multiserver Queueing
Systems

• Lets compare the mean delay for
• a service center having a single server with
  service rate ? customers per minute, and
• a service center having 2 servers each with
  service rate ?/2 customers per minute.

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Comparing Single Server and Multiserver Queueing
Systems

• We can readily show that

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Comparing Single Server and Multiserver Queueing
Systems

• Example ? 10 customers/minute

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Comparing Single Server and Multiserver Queueing
Systems

• In general for Markovian queues,

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M/G/1 Queue

• M/G/1 queueing system
• exponential interarrival distribution with rate ?
• general service distribution with mean t and
  variance s 2
• 1 server
• infinite queue
• arrivals are independent of the state of the
  system
• FCFS service discipline

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M/G/1 Queue

• The random process N(t) (the number of customers
  in the system at time t) is no longer a Markov
  process. (Why?)

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M/G/1 Queue

• If X0(t) is the service time already received by
  the customer in service at time t, then the
  process (N(t), X0(t)) is a Markov process. (Why?)
• However X0(t) has a continuous state space, so we
  cannot apply the tools we have developed for
  discrete-valued Markov processes.
• We need some new tools to analyze this system.
  One of these is the method of the imbedded Markov
  chain.

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Embedded Markov Chain

• With the method of the embedded Markov chain, we
  consider the state of the system only at a select
  set of points in time.
• For the M/G/1 queue, we choose to consider the
  system only at the times of a departure from the
  system.
• What is the age of the service of the customer in
  service at the time of a departure?
• Assuming that t is the time of a departure, the
  state at time t is now completely described by
  the number of customers in the system, N(t) (a
  discrete random variable).
• The times of departures form a Markov chain.

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Embedded Markov Chain
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Equality of the Departing Customers and the
Arriving Customers Distributions

• Theorem 1. Consider a queueing system for which
  the realizations of the state process are step
  functions with only unit jumps (positive and
  negative). Then the equilibrium state
  distribution just prior to an arrival is the same
  as the equilibrium state distribution just
  following a departure, i.e., arriving customers
  and departing customers see the same probability
  distribution for the state of the system.

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Equality of the Departing Customers and the
Arriving Customers Distributions
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M/G/1 Queue

• Recall that for an M/M/1 queue in equilibrium,
  the expected number of customers in the queue and
  the expected delay in the queue waiting for
  service are given by
• where ? ?t.
• What are these quantities for an M/G/1 queue?

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Pollaczek-Khintchine Formula

• For an M/G/1 queue in equilibrium, the expected
  number of customers in the queue and the expected
  delay in the queue waiting for service are given
  by
• The latter identity is the Pollaczek-Khintchine
  formula.

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Pollaczek-Khintchine Formula

• Proof of the P-K formula Let
• Nk N(tk)
• the number of customers left behind by the
• k th departing customer
• Xk the number of customers that arrived during
  the
• service epoch of the k th departing customer

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Pollaczek-Khintchine Formula

• If the k th departing customer does not leave the
  system empty, then Nk gt 0 and
• If the k th departing customer leaves the system
  empty, then Nk 0 and

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Pollaczek-Khintchine Formula

• Both these equations can be combined into the
  single equation
• where
• Note that
• and

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Pollaczek-Khintchine Formula

• Squaring both sides of (1) gives

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Pollaczek-Khintchine Formula

• Taking expected values, we get
• Now let k ? ?, and assume the system is in
  equilibrium. Then
• It follows that if X and N refer to the system at
  the time of a departure, then

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Pollaczek-Khintchine Formula

• Returning to (1)
• and taking limits and expectations gives
• and

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Pollaczek-Khintchine Formula

• It only remains to calculate EX and EX 2.
  Recall that X is the number of arrivals during an
  arbitrary service time. Since the arrival process
  is exponential, the number of arrivals in a time
  period t is Poisson with parameter ?t. It follows
  that
• so that

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Pollaczek-Khintchine Formula

• Putting this together with (2) gives
• Note that this is comprised of two parts the
  expected number in the server (?) and the
  expected number in the queue

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Pollaczek-Khintchine Formula

• Finally, by Littles Theorem, the expected delay
  is given by
• and

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Pollaczek-Khintchine Formula

• We have established these formulas only for the
  times of a departure. However, by Theorem 1,
  these formulas also apply for the system as seen
  by an arriving customer.
• Since the arrivals are Poisson, the formulas also
  describe the system as seen by an outside
  observer, i.e., they hold for all time t.

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Comparing Service Distributions

• Example t 1

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Comparing Service Distributions
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Busy Periods in an M/G/1 Queue

• The busy period is the length of time from the
  instant that the (previously idle) server begins
  serving a customer until the server next becomes
  idle and no customers are waiting in the queue.

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Busy Periods in an M/G/1 Queue

• In general, the busy period is difficult to
  analyze, but the mean busy period can readily be
  calculated for a queue with exponential arrivals.
• Let b be the mean length of the busy period. Then
  we have
• Solving for b yields

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Network of Queues

• Let us now consider a set of interconnected
  queues, possibly with feedback
• We restrict our attention to an open network of
  queues, i.e., a network with external arrivals.

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Laplace Transforms Revisited

• Recall that for a continuous non-negative R.V. X
  with probability density function g(x), its
  Laplace transform is given by

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Laplace Transforms Revisited

• Facts about Laplace transforms
• The Laplace transform ? of an exponential R.V.
  with rate v is given by
• The Laplace transform of the sum of two R.V.s is
  the product of their Laplace transforms.
• A R.V. is uniquely determined by its Laplace
  transform.

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Output of a M/M/1 Queue

• Theorem 2. Consider and M/M/1 queue with arrival
  rate ? and service rate ?, such that ? lt ?.
• Then the departure intervals are independently
  distributed with distribution function

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Output of a M/M/1 Queue

• Proof Let the random variable U denote the time
  interval between two successive departures with
  distribution function D and Laplace transform d
  also let A denote the interarrival time and S the
  service time. Now consider a departing customer.
  With probability ? the departing customer will
  not leave the system in an idle state in this
  case, U S. With probability 1 ? ?, the
  departing customer leaves behind an empty system,
  so that U A S.

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Output of a M/M/1 Queue

• It follows that
• In terms of Laplace transforms, we have that

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Queues in Tandem

• Since the output process of each queue is Poisson
  with rate ?, then the input of each queue is
  Poisson, and each queue can be analyzed
  separately as an M/M/1 queue.

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Burkes Theorem

• More generally, the steady-state output of a
  stable M/M/s queue with arrival rate ? and
  service rate ? for each of the s servers is a
  Poisson process with rate ?, and is independent
  of the arrival and service processes.

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Jacksons Theorem

• In an open network of queues with Poisson
  arrivals, FCFS queues with exponential service
  times and no saturated queues,
• Each individual queue may be treated as an M/M/1
  queue with arrival rate equal to the throughput
  to obtain its queue length.
• In a network of M queues, the joint queue length
  distribution of the network is the product of the
  queue length distributions of each queue, i.e.,

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Homework

• Suppose that a single server queue does not have
  a Poisson process input and does not have an
  exponential server. What variables are necessary
  to describe the state of the system at any given
  instant of time?
• A small business has three outside lines for its
  telephones. Measurements show that the average
  call lasts 1 minute and 2 calls are generated per
  minute. The current probability of not getting an
  outside line is 0.21, which is considered
  unacceptable. How many additional outside lines
  should be added to get the blocking probability
  below 5? Note that blocked calls are lost.

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Homework

• Is the following table realizable for a finite
  buffer state-independent queueing system? Why or
  why not?

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Homework

• Consider a 3-queue tandem Markovian network with
  arrival rate ? 10 customers per second and
  service rates ?1 12 customers per second, ?2
  15 customers per second, and ?3 20 customers
  per second. What is the average number of
  customers in the network? What is the average
  delay through the network?
• Ross 8.19

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References

• Erhan Cinlar, Introduction to Stochastic
  Processes, Prentice-Hall, Inc., 1975.
• Robert B. Cooper, Introduction to Queueing
  Theory, Second Edition, North Holland, 1981.
• Leonard Kleinrock, Queueing Systems, Volume I
  Theory, John Wiley Sons, 1975.
• Sheldon M. Ross, Introduction to Probability
  Models, Ninth Edition, Elsevier Inc., 2007.

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References

• Richard W. Conway, William L. Maxwell, Louis W.
  Miller, Theory of Scheduling, Dover Publications
  Inc., 1967.
• Charles H. Sauer, K. Mani Chandy, Computer
  Systems Performance Modeling, Prentice Hall,
  Inc.,1981.
• Thomas G. Robertazzi, Computer Networks and
  Systems Queueing Theory and Performance
  Evaluation, Third Edition, Springer-Verlag, 2000.