Retrial Queues with Losses: A Martingale Approach

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Retrial Queues with LossesA Martingale Approach

• Guichang Zhang and Raj Srinivasan
• University of Saskatchewan
• Aug 27, 2008
• Zhangg raj_at_math.usask.ca

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Outline of the talk

• Introduction
  
• Assumptions
• Main methods and results
  
• The counting processes
• Flow equations
• Main result
• Two simple cases

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The sketch of the retrial queue
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Assumptions on customers flow

• An arriving customer enters the service process
  if there is at least one free server, after a
  random service time he will leave the system
• If all the servers are busy, the customer will
  enter the orbit if there is at least one free
  space in the orbit. Otherwise, the customer will
  be lost
• For the customers in the orbit, they will retry
  to occupy a server after some random time. The
  retrying customer will occupy one server if there
  is at least one free server. Otherwise, the
  customer will enter the orbit and retry again.

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Other assumptions

• The arrival process, departure processes and
  retrial processes are independent of each other
  and have no common jump almost surely for each t
• Respectively, the arrival time, retrial time and
  service time are generally distributed which are
  strictly stationary and ergodic.

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Main methods and results

• Martingale method. Especially Doob-Meyer type
  decomposition theorem
• Indicator function method
• Under the general assumptions, we provide a weak
  sense of equilibrium for the retrial queue system.

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The structure of counting processes

• In probability space
  , let denote
• the random time between two consecutively
  arriving
• customers.

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• Let

Then the arriving counting process is
From Doob-Meyer decomposition theorem
Where is F-martingale and is the
compensator of A which can be written as
Where is adapted to
the filtration F and is called the stochastic
intensity of the counting process A.
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• For every server, there is a potential departure
• process. denotes the random service time.

With this method, we construct n independent
potential counting processes for each server
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• For every orbit space, there is a potential
  retrial
• process. denotes the random retrial time.

With this method, we construct m independent
potential counting processes for each orbit
space
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Flow equations of the system
(1)
(2)
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Q2
Equation 2
Q3
1
2
m
1
2
n
D
Q1
A
Equation 1
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• From Doob-Meyer decomposition theorem

Where M1,M2 and M3 are local square integrable
Martingales with
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The expectation equations
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• In classical situations, one looks at

Here we consider the following limits (in a
weaker sense)
For the existence, please refer to Abramov. V.
M. Abramov, Analysis of multiserver retrial
queueing system A martingale approach and an
algorithm of solution, Ann. Oper. Res. 141(2006)
19-50.
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Main result

• Theorem 4.1 Given three independent counting
  processes
• A, Di and Rj with stochastic intensities X, Y
  and Z
• respectively, and having strictly stationary and
  ergodic
• increments, the retrial queue system with losses
  is driven
• by them. Let Q1 and Q2 denote the number of
  customers in
• the queue and orbit, respectively. Let
• where IA is the indicator of event A. Then we
  have the
• following relationships.

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For
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Remark

• For general case, there are three states (i-1,j),
  (i-1,j1) and
• (i1,j) which can transit into state (i,j) and
  three states
• (i1,j), (i-1,j) and (i1,j-1) which come from
  state (i,j) .

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The equilibrium in a weaker sense

• The left hand side in the equations means the
  Cesaro limit
• of rate into state (i,j) the other side means
  the Cesaro limit
• of rate out from state (i,j). The main result
  shows there is
• an equilibrium in a weaker sense in our retrial
  queue
• system when time goes to infinity.

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Denote
In the following, we will give two special cases.
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Special casenonhomogeneous Poisson Processes

• Corollary 4.2 Assume the counting processes A,
  Di and Rj
• in theorem 4.1 are nonhomogeneous Poisson
  processes,
• then we have the following equations

For
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Special casehomogeneous Poisson Processes
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Thank you !