Two Problems in Stochastic Service Systems

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Two Problems inStochastic Service Systems

• Jun Guo
• Supervisors Moshe Zukerman, Peter Taylor, Hai Le
  Vu

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Outline

• Stochastic service system
• Part 1 Delay analysis of finite buffer TDMA
  systems
• Constant service
• State-dependent service
• Part 2 File assignment optimization for VOD
  systems
• Performance analysis
• Heuristic allocation method
• Problem transformation
• Single-objective optimization
• Multi-objective optimization

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Stochastic Service System

• Random arrival service ? contention for finite
  system resources
• Delay Blocking, given a finite buffer
• Blocking, given no buffer
• Two typical performance measures
• Waiting time distribution
• Probability of a packet waiting longer than 20
  sec in the buffer should not exceed 5.
• Blocking probability
• Probability of a user request being blocked
  should not exceed 1.

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Time Division Multiple Access (TDMA)

• Unique time slots for each user to transmit data
  packets across the shared channel, no
  interference
• Wide applications in telecommunications systems
  and computer communication systems GSM, GPRS,
  EDGE, PDC, iDEN,
• Few results in the literature on delay analysis
  for finite buffer TDMA systems, all for constant
  service
• Constant unit service in Birdsall62, flawed
• Constant batch service in Ryden93 and
  Simonot95, rather complicated

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Assumptions

• Time axis divided equally into successive time
  slots of length T
• Homogeneous Poisson packet arrival process with
  rate ?. The probability of n
  arrivals during time t is
• Size of the buffer K packets.
• An arriving packet is admitted if fewer than K
  packets are present in the buffer, otherwise it
  is lost.
• Service occurs at the end of a time slot.
• First come first served

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Embedded Markov Chain

• Let Jk be the number of packets in the buffer at
  time kT. Jk is a Markov chain with state space
  0, 1, , K.
• The probability transition matrix
  is given by
• are the
  steady-state probabilities of Jk.

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Waiting Time

• A packet admitted in (0, T) may be removed at
  kT?, k 1. If it is admitted at time T?u, 0 lt u
  lt T, its waiting time is then u(k?1)T.

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8
4
6
7
3
2
1
3
Remove 2
Remove 1
Remove 3
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Constant Unit Service

• Given at least one packet in the buffer at time
  kT?, one and only one packet is removed from the
  buffer.
• Waiting time bounded by KT.

Waiting Time Density
Wrong!
K 1
?T 0.8
Value of Density
K 2
K 3
Time
Results from Birdsall62
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Why Birdsall62 is wrong?

• If the size K of the buffer is one, and the
  buffer is empty at time kT, only the first
  arriving packet can be admitted.
• This packet is removed from the buffer at time
  (k1)T?.
• The waiting time of an arbitrary admitted packet
  is with emphasis toward T.

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Our Solution

• expected number of packets admitted during
  (0, T)
• expected number of packets
  admitted during
• (0, T) whose delay lies between u and udu
• waiting time density of an arbitrary
  admitted packet

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for Constant Unit Service

• Waiting time bounded by KT.
• For an arbitrary admitted packet to depart at
  time kTT,
• 0 k K-1, it must arrive and occupy exactly
  the (k1)th position in the buffer.
• On the interval (kT, kTT)

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Our Results
Waiting Time Density Histogram
?T 0.8
K 1
Value of Density
K 2
K 3
Time
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Constant Batch Service

• Given N or more packets in the buffer at time
  kT?, N packets are removed otherwise, all
  packets in the buffer will be removed.
• Let , waiting time
  bounded by MT
• Solutions in Ryden93 and Simonot95 rather
  complicated, no rigorous proof of validity

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for Constant Batch Service

• For an arbitrary admitted packet to depart at
  time kTT, 0 k M-1, it must arrive and
  occupy one of the positions r, Nk1 r NkN,
  in the buffer.
• On the interval (kT, kTT), 0 k M-3
• On the interval (MT-2T, MT-T)
• On the interval (MT-T, MT)

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Results for Constant Batch Service
K 70, N 10, ?T 10
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State-Dependent Service

• Motivated from GSM paging in Ivanovich03
• Given j packets in the buffer at the end of a
  time slot, remove i of j with conditional
  probability , and
• Versatility
• Generalize constant service
• Support delay analysis for time-slotted optical
  burst switching networks Vu05
• Quality-of-service classification, demand
  assigned
• If , waiting time arbitrarily
  large with positive probability

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for State-Dependent Service

• We must keep track of the buffer content at each
  epoch kT and of the position r, K r 1, of
  the packet that we are following.
• A matrix formalism to handle the complex
  accounting of the queueing process
• Expressions for in this case were
  presented in my earlier progress talk on 03 Dec
  2003.

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Other Results
Little Formula

• Mean waiting time
• is the limiting time-average number of
  admitted packets in the buffer.
• is the limiting time-average rate at
  which packets are admitted.
• Blocking probability

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Video-on-Demand (VOD)
Limited storage space limited bandwidth
Large file-size large bit-rate
Long-lived connection
Blocked
User Community
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File Assignment Problem

• Given a large number of disks, with limited
  capacity in both storage space and I/O bandwidth,
  and a large library of movie titles, with
  significant asymmetry in access demand and
  file-size, how to assign movie files to disks, so
  that the blocking probability of user requests is
  minimized, subject to capacity constraints.

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Notation
Set of disks in the system Number of disks in the
system Disk storage space (in unit) Set of movies
in the system Number of movies in the
system File-size (in unit) of movie m Number of
file-copies of movie m Popularity profile of
movie m Mean request arrival rate (Poisson) Mean
movie connection time
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Mathematical Formulation
Request Blocking Probability

• Minimize
• Such that

Movie Availability Constraint
Disk Storage Space Constraint
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Earlier Attempts

• Little95, Mourad96, Serpanos98, Tang01
  and Leung05 assumed an inefficient resource
  selection scheme Single Random Trial (SRT)
  Single random selection among the set of disks,
  no subsequent retrials attempted
• Easy to analyze RBP ?
• Underutilize system resources given the existence
  of multi-copy movies ?
• Wolf97, Tsao99, and Zhao03 assumed all
  movies of identical file size
• Using a so-called apportionment method to decide
  number of file-copies for each movie title ?
• Unrealistic ?

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Our Assumptions

• We assume an efficient resource selection scheme
  Least Busy Fit (LBF) Select the least busy disk
  among the set of disks
• Utilize system resources efficiently ?
• State-dependent traffic, difficult to analyze RBP
  ?
• We assume movies of different file sizes
• Realistic ?
• No straightforward method to decide number of
  file-copies for each movie title ?
• This file assignment problem is more realistic
  but more challenging.

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Performance Analysis of LBF

• Difficulty of analysis
• Multidimensional Markov process (dimension J )
• Curse of dimensionality, computationally
  infeasible
• Developed approximate solutions to evaluate RBP
  of LBF using the well-known fixed-point
  approximation method Kelly86
• Decouple the whole system of J disks into J
  independent subsystems
• Analyze each subsystem using Erlang-B formula
• Fast and sufficiently accurately differentiate
  performance of different file assignments ?
• Limited information about RBP ?

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Heuristic Allocation Method

• Type c movies All movies that have c file-copies
• For a given replication instance
  , an ideal allocation instance is,
  for each c, the traffic wishing to access movies
  of Type c is uniformly distributed among all
  combination groups enumerated in the set of all J
  disks in the system ? Combination Load Balancing
  (CLB)
• At the state of CLB, RBP is minimal for the given
  replication instance.
• CLB motivates our design of a good performance
  heuristic allocation method, deterministic ?

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Problem Transformation

• Divide the entire solution space into subspaces
• All file assignment solutions in a subspace ?
• A common replication instance
• Heuristic allocation instance ? The local
  optimal solution of each replication instance
• Original problem file assignments ?
• Transformed problem replication
  instances
• Further, for a given replication instance, if
  ,
• no file assignment solutions ? strictly
  non-allocatable
• Interestingly, even if ,
  not strictly allocatable ? likely allocatable

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Example

• 3 disks, 4, 12
• 8 movies,
• 1.17, 0.67, 1.08, 1.07, 1.25, 0.61,
  0.77, 1.00
• Original problem 224 16,777,216 file
  assignments ?
• Transformed problem 38 6,561 replication
  instances
• Further, only 507 likely allocatable replication
  instances
• In fact, only 355 strictly allocatable
  replication instances
• Optimal solution
• Original problem LBF RBP 0.00302
• Transformed problem LBF RBP 0.00422
• Drastically reduced and yet quite effective
  solution space

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Genetic Algorithm (GA)

• Inspired from the mechanism of natural selection,
  survival of the fittest
• Population-based stochastic search optimization
• Start from an initial population of randomly
  generated solutions
• Perform multi-directional stochastic search
  through a genetic evolution process
• Selection, crossover, mutation, replacement
• Only rely on the objective function, no auxiliary
  knowledge required ?

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How GA is used in our context?
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Single-Objective Optimization

• RBP as the objective in each cycle of the
  evolution process

20 disks, 200 movies
0.958
0.792
LBF RBP ()
0.252
(a) (b)
(c)
(a) Single-objective GA using LBF, CPU time 204
min (b) Random search (c) Single-objective GA
using SRT
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Performance Indices
MTI
STI

• MTI quality of assignment on multi-copy movie
  files
• STI quality of assignment on single-copy movie
  files
• Two conflicting objectives

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Justification of Conflicting Relationship
3 disks, 8 movies
MTI
STI
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Multi-Objective Optimization

• Promote RBP as higher level decision maker
• Use MTI and STI as the conflicting objectives in
  each cycle of the evolution process

20 disks, 200 movies
LBF RBP ()
0.252
0.247
(a)
(b)
(a) Single-objective GA, CPU time 204 min (b)
Multi-objective GA, CPU time 50 min
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Non-dominated front in the last generation

• Authentic conflicting relationship between MTI
  and STI

20 disks, 200 movies
LBF RBP ()
Minimal RBP
MTI
STI
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Summary

• Presented delay analysis of a new finite buffer
  TDMA model with state-dependent stochastic
  service
• Provided concise treatments for delay analysis of
  traditional finite buffer TDMA models with
  constant unit service or constant batch service
• Presented a viable proposal using evolutionary
  algorithms to solve a file assignment problem for
  a large scale VOD service system computationally
  efficiently