GPRS Optimization of GPRS Time Slot Allocation Considering Call Blocking Probability Constraints

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• ????????GPRS???????? Optimization of GPRS Time
  Slot Allocation Considering Call Blocking
  Probability Constraints

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Outline

• Introduction
• Background
• Motivation
• Literature Survey
• Proposed Approaches
• Discrete-time Model
• Problem Formulation
• Computational Results
• Continuous-time Model
• Problem Formulation
• Computational Results
• Future Research

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Background

• Wireless network and mobile applications have
  become more and more popular.
• Mobile data services are more widely deployed and
  accepted by users than before.
• GPRS is a better data solution than GSM before
  3G.
• By using the dynamic slots allocation.
• Can be viewed as a gangway to 3G.
• The need of the QoS in GPRS is an essential
  issue.
• There are still many research issues regarding
  the end-to-end QoS in GPRS networks.

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Motivation

• GPRS and GSM share the same radio resources.
• Since the channels are shared by data and voice
  users and can be dynamically allocated to GPRS
  data users, the allocation policy plays an
  important role of the system revenue and the QoS.
• The voice channel assignment should be considered
  with QoS concerns, e.g. call blocking.
• Our goal is to find a best slot allocation policy
  
• to maximize the total system revenue.
• subject to the QoS constraint.

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Literature Survey

• GPRS is based on the GSM platform.
• New radio channels are defined in GPRS using GSM
  Architecture.
• The allocation of these radio channels is
  flexible.
• One to eight time slots can be allocated to a
  user.
• Several active users can share a single time
  slot.
• Each TDMA frame has 8 time slots.

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Markovian Decision Process

• MDP is an application of dynamic programming.
• It is used to solve a stochastic decision process
  that can be described by a finite number of
  states.
• The transition probabilities between the states
  and the reward structure of the process resulting
  from moving from one state to another are
  described by a Markov chain.
• Both the transition and revenue matrices depend
  on the decision alternatives available to the
  decision maker.
• The objective of the problem is to determine the
  optimal policy that maximizes the expected
  revenue of the process over a finite number of
  states

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Proposed Approach

• We introduce three approaches to address the call
  blocking probability constraints in the maximum
  system revenue model.
• Linear programming.
• Lagrangian relaxation.
• Expansion of the Markovian Decision Process (MDP)
  model.

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Linear Programming

• The finite-state Markovian decision process can
  be formulated and solved as a linear program.
• We will add the call blocking probability
  constrains to the original model and solve it by
  the linear programming.
• The optimal solution may be infeasible, e.g. a
  fractional number.
• Rounding heuristics will be developed.

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Lagrangian Relaxation

• We regard the maximum system revenue model as a
  macro-view model.
• Add the call blocking probability constraints to
  it.
• Change the system revenue by relaxing the call
  blocking probability constrains.
• Then use the new system revenue in the Markovian
  decision process and solve the maximum system
  revenue model.

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Expansion of the MDP Model

• The optimal policy can not be found directly if
  we consider the call blocking probability in the
  Markovian decision process.
• What we can do is to use heuristics to find
  nearly optimal solutions.
• The sequence of states to be changed
• The decision of each state that must be arranged

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Discrete-time Model

• Our assumptions are
• Both voice and data messages arrivals follow a
  Geometric distribution.
• We give the probability p for users in system to
  calculate the leaving probability.
• Single-slot data users and two-slot data users
  are independent.
• The 1/p of each traffic is the average holding
  time of users.
• For each voice users, only one slot is dedicated.
  
• For data users, one or two slots are dedicated.
• A slot is called a channel.
• There are total 8 channels in our model.
• There will be multiple users arrival or
  departure in each time interval.

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Problem Description

• To determine
• The best slot allocation policy.
• Objective
• To maximize the system revenue.
• Subject to
• Slot capacity constrains.
• QoS constrains
• An upper bound on the call blocking probability
  for each class of traffic, e.g., voice users,
  single-slot data users and two-slot data users.

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Problem Description (contd)

• We define the system state as (a, b, c)
• a means the number of voice users
• b means the number of single-slot data users
• c means the number of two-slot data users

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3-Dimensional Markov Chain Model (discrete-time)
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Details of a Triangle (discrete time)
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Notation Given Parameter
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Notation Given Parameter (contd)
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Notation Decision Variables
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Notation Decision Variables (contd)
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Linear Programming Formulation
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Linear Programming Formulation (contd)
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Lagrangian Relaxation Formulation
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Lagrangian Relaxation Formulation (contd)
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Lagrangian Relaxation Formulation (contd)

• Sub problem

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Expansion of the MDP Formulation
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Expansion of the MDP Formulation (contd)
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Solution Procedure
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Solution Procedure (contd)

• Compared model (both for discrete and continuous
  case)
• We construct one fixed channel with 5 channels
  assigned to the voice users to guarantee the call
  blocking probability of 0.01.
• The remaining 3 channels will be assigned to the
  data channels with the smaller size model the
  same as the Markovian decision process model in
  our thesis.
• We will view the two data channels as the two
  dimensional vector as (x, y). And the capacity
  constraint will be x2y?3.
• The total states of the 3-channel Markovian
  decision process will have 6 states only.
• We will call this model as the compared model.

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Solution Procedure (contd)

• We will give the parameters
• the arrival rate probability, service rate
  probability, and the revenue to each type of
  traffics.
• We use the information to calculate the
  probabilities of transitions of each valid pair
  of states and the corresponding revenue.
• Then we will solve this problem using
• The simplex method of linear programming.
• Lagrangian relaxation.
• Expansion of the MDP.
• We will not set the call blocking probability
  upper bound of all three types of traffics
  tightly.
• In this thesis, we will focus on the voice users
  call blocking probabilities.

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Linear Programming Arrangement

• If the problem solved by the linear programming
  results in a non total integer solution, we will
  arrange the states that contribute to the
  fraction solution.
• We will calculate the fractions and the
  corresponding expected revenue.
• Find out the maximum one that will less infect
  the solution of system revenue and will not
  conflict to our call blocking probability
  constraints.

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Linear Programming Arrangement Example

• For example, if the solution of state i results
  in
• one decision 1 with fraction 0.8, expected
  revenue 12.
• the other decision 4 with fraction 0.2, expected
  revenue 8.
• we will compare the two arrangement of
• Choosing the decision 1 will result in the change
  of the total system revenue of 0.2 12 0.2
  8.
• Choosing the decision 4 will result in the change
  of the total system revenue of 0.8 8 0.8
  12.
• Then we will calculate the states probabilities
  and find out the one that satisfies our call
  blocking probability constrains or the nearly
  satisfied one.

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Linear Programming Arrangement Results

• This result shows that when we make an
  arrangement of the linear programming solution,
  we will have to make decision between the
  trade-off of the system revenue and the call
  blocking probability.
• When we choose to increase the system revenue,
  the call blocking probability will decrease, and
  vise versa.
• If the call blocking probability increase under
  1, we will accept this result.

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Expansion of the MDP Model Heuristics
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Expansion of the MDP Model Heuristics (contd)
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Computational Results

• The increase of the system revenue and the
  difference between our three models and the
  compared model occurs
• with the increase of the average holding time of
  the traffics.
• with the higher revenue of the traffics.
• When the service rate and the arrival holding
  time can match with each other, all of the three
  approaches can almost reach the same total
  revenue under the call blocking probability
  constraints.
• For extreme cases, some problems can not be
  solved by EMDP.
• The benefit of our approaches is almost twice as
  the compared model.

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Computational Results (contd)

• The light blue line represents the result of the
  compared model.
• Figure1
• The blue line represents the result of the call
  blocking upper bound is 1, where the green line
  is 0.05 and the red line is 0.01.
• Both voice users duration time and data users
  duration time decrease from left to right.
• Figure2
• The blue line means that the problem is solved by
  the linear programming, where the green line is
  by the Lagrangian relaxation and the red line is
  by the EMDP.
• Voice users duration time is the same.
• Data users duration time increases from left to
  right.
• The call blocking probability upper bound is
  0.01.
• Figure3
• An extreme case. The problem can not be solved by
  EMDP.

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Computational Results (contd)

• Figure 1 of the discrete-time case.

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Computational Results (contd)

• Figure 2 of the discrete-time case.

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Computational Results (contd)

• Figure 3 of the discrete-time case.

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Conclusion

• The linear programming approach costs the most
  time.
• Next is the Lagrangian relaxation and the
  Expansion of MDP costs the least.
• We can conclude that the Lagrangian relaxation is
  the most suitable approach for the total
  scalability of the discrete time case.

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Continuous-time Model

• Our assumptions in continuous model
• Both voice and data messages arrival follow a
  Poisson process and the call holding time and
  data message length are exponentially distributed
  with different parameters.
• Single-slot data users and two-slot data users
  are independent.
• For each voice users, only one slot is dedicated.
  For data users, one or two slots are dedicated.
• A slot is called a channel.
• There are in total 8 channels in our model

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Problem Description

• To determine
• The best slot allocation policy.
• Objective
• To maximize the system revenue.
• Subject to
• Slot capacity constrains.
• QoS constrains
• An upper bound on the call blocking probability
  for each class of traffic, e.g., voice users,
  single-slot data users and two-slot data users.

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Problem Description (contd)

• We define the system state as (a, b, c)
• a means the number of voice users
• b means the number of single-slot data users
• c means the number of two-slot data users

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3-Dimensional Markov Chain Model
(continuous-time)
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Details of a Triangle (continuous time)
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Notation Given Parameter
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Notation Given Parameter (contd)
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Notation Decision Variables
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Linear Programming

• Because there is no proper continuous-time case
  model in the Markovian decision process using the
  linear programming in Operational Research.
• We construct one model like the discrete-time
  case and use a to represent the
  transition probability.
• When the is small enough, the probability
  of the transition that has more than one change
  will become very small.
• The discrete-time case model can be viewed
  similar as the continuous-time case model.

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Lagrangian Relaxation Formulation
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Lagrangian Relaxation Formulation (cont')
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Lagrangian Relaxation Formulation (cont')

• Sub problem

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Expansion of MDP Formulation
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Expansion of MDP Formulation (contd)
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Solution Procedure
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Solution Procedure (contd)

• We will give the parameters
• arrival rate, service rate, and the revenue to
  each type of traffics.
• We use the information to calculate the rates of
  transitions of each valid pair of states and the
  corresponding revenue.
• Then we will solve this problem using
• Lagrangian relaxation.
• Expansion of the MDP.
• We will not set the call blocking probability
  upper bound of both types of traffics strictly.
• In this thesis, we will focus on the voice users
  call blocking probabilities.

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Computational Results

• The difference between our two models and the
  compared model becomes larger.
• With the increase of the arrival rates of the
  three traffics.
• With the increase of the revenue of the three
  traffics.
• The larger the arrival rate
• the larger the total system revenue.
• the larger the difference between our two models
  and the compared model.

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Computational Results (contd)

• The red lines represents the result of the
  compared model
• Figure1
• The blue line means that the problem is solved by
  the Lagrangian relaxation, where the green line
  is by the EMDP.
• Voice and Data users arrival rates decrease from
  left to right.
• The call blocking probability upper bound is
  0.01.
• Figure2
• A general case.
• Upper small figures the left one is the result
  solved by the Lagrangian relaxation, and the
  right one is the result solved by EMDP.
• Bottom small figures the comparison of the
  Lagrangian and the EMDP with call blocking upper
  bound of 0.05 and 0.01.
• Figure3
• An extreme case.
• Some sample of this problem can not be solved by
  EMDP.

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Computational Results (contd)

• Figure 1 of the continuous case.

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Computational Results (contd)

• Figure 2 of the continuous case.

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Computational Results (contd)

• Figure 3 of the continuous case.

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Conclusion

• The results of the Lagrangian relaxation approach
  and the expansion Markovian decision process are
  almost the same, except for the extreme case in
  which some sample can not be solve by EMDP.
• The system revenue of our two models is almost
  twice as the system revenue of the compared
  model.
• The computational time of Lagrangian relaxation
  approach is almost 1.76 times more than the
  expansion of the Markovian decision process
  approach.
• We conclude that in the continuous time case, the
  Lagrangian relaxation approach is the better
  approach for large scalability.

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Future Research

• More channels.
• More types of traffics.
• More QoS constraints.

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The End

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