The Waiting Time Distribution for a TDMA Model with a Finite Buffer and State Dependent Service

1
The Waiting Time Distribution for a TDMA Model
with a Finite Buffer and State Dependent Service

• Marcel F. Neuts, Jun Guo, Moshe Zukerman and Hai
  Le Vu

2
Objective
Derive exact steady-state waiting time
distribution of an arbitrary admitted item
(message/packet) in a finite buffer TDMA system
with state dependent service.
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Motivation
T D M A

• Wide applicability
• Important module in various large systems
  Multi-access, WDM Edge router,
• First attempt in 1962 by T. G. Birdsall, but it
  was flawed

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Whats Coming

• Finite buffer TDMA model with state dependent
  service
• Exact waiting time distribution of an arbitrary
  admitted item in the system
• Mean waiting time of an arbitrary admitted item
  in the system, using Little formula
• Future work

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GSM Paging State Dependent Service

• The service rate of each paging sub-channel is 2,
  3 or 4 paging messages per T, depending on the
  type of messages waiting in the buffer.

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The Finite Buffer TDMA Model
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Waiting Time

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

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8
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2
1
3
Remove 2
Remove 1
Remove 3
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Asynchronous Time Multiplexing Constant Service

• For , if
• our finite buffer TDMA model then reduces to the
  one in Birdsall 1962.
• Given at least one item in the buffer at time
  kT?, one and only one item is removed from the
  buffer.

Reference Birdsall et al., Analysis of
asynchronous time multiplexing of speech sources,
IRE Trans. Commun. Syst., 1962.
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Appetizer Birdsalls Results
Wrong ! Why ?
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Why Birdsalls Results Are Wrong?

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

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Our Results
Waiting Time Density Histogram
?T 0.8
K 1
Value of Density
K 2
K 3
Time
Radii of 95 confidence intervals lt 0.02
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Milestone

• Finite buffer TDMA model with state dependent
  service
• Exact waiting time distribution of an arbitrary
  admitted item in the system
• Mean waiting time of an arbitrary admitted item
  in the system, using Little formula
• Future work

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Derivation Outline

• Let Jk be the number of items in the buffer at
  time kT. Jk is a Markov chain with state space
  0, 1, , K.
• By , we denote the
  steady-state probabilities of Jk.
• Let E be the expected number of items admitted
  during a slot of length T.
• is the expected number of items
  admitted during (0, T) whose delay lie between u
  and udu.

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Derivation Outline (cont.)

• Let be the probability density of the
  delay distribution of an arbitrary admitted item,

• We must keep track of the buffer content at each
  epoch kT and of the position r, K r 1, of
  the item that we are following.
• When the tracked item leaves the buffer, we say
  that it reaches position 0.

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Accounting for the First Time Frame .
after service
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Derivation of .

• The quantity is
  positive only when r r' and i' r ? r' K.
  Note that i' r ? r' is the number of items in
  the buffer immediately before service.
• For i' r ? r' lt K, the buffer is not filled up
  during (T?u, T), and the expression of
  for that case is PA(u)
• For i' r ? r' K, the buffer is filled up
  during (T?u, T), and the corresponding expression
  of is PB(u)

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The Structure of .
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Accounting for the First Time Frame .

• Define the column vector of dimension
  K(K1)/2, for K r 1 and i r ? 1.
• The quantity is the
  elementary conditional probability of the
  depicted event.

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Accounting for Subsequent Time Frames .
after service
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Derivation of .

• The quantity is positive
  only when r1 r2, i1 ? i2 r1 ? r2, and i2
  r1? r2 K.
• For i2 r1 ? r2 lt K, the buffer is not filled up
  during (kT, kTT), and the expression of
  for that case is PC
• For i2 r1 ? r2 K, the buffer is filled up
  during (kT, kTT), and the corresponding
  expression of is
  PD

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The Structure of .
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Accounting for Subsequent Time Frames .
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The Stochastic Matrix .
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Derivation of .
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Waiting Time Density

• is a valid probability density.
• To improve the computation efficiency,
  can be nicely unpacked into a finite, positive
  linear combination of gamma densities and beta
  densities on each of the successive intervals
  (kT, kTT), for k 0.

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Delay Distribution

• One way to calculate the mean waiting time is

• The variance of the delay is calculated by

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Simulation Study

• Six independent runs, each of which consists of
  one million independent time slots.
• In each time slot, regenerate the Poisson process
  until at least one item arrives.
• Keep track of the waiting time of each of those
  admitted items until they are removed from the
  buffer.
• Update the waiting time density histogram, from
  which the empirical delay distribution is
  estimated.

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Numerical Example

• Consider a buffer of size K 70, under loads (?T
  1, 60, 600 ).
• The parameters were specified as
  follows
• For j even, we formed the j1 integers 1, 2, ,
  j/2, j/21, j/2, , 2, 1, divided by their sum.
  For j odd, we formed the j1 integers 1, 2, ,
  ( j1)/2, ( j1)/2, , 2, 1, divided
  by their sum, and so identified .

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Numerical Results (?T 1)
Radii of 95 confidence intervals lt 2.6
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Numerical Results (?T 60)
Radii of 95 confidence intervals lt 1.5
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Numerical Results (?T 600)
Radii of 95 confidence intervals lt 1.1
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CPU Time Distribution
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Milestone

• Finite buffer TDMA model with state dependent
  service
• Exact waiting time distribution of an arbitrary
  admitted item in the system
• Mean waiting time of an arbitrary admitted item
  in the system, using Little formula
• Future work

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Little Formula L ?W

• L is the limiting time-average number of
  customers in the system.
• ? is the limiting time-average arrival rate.
• W is the limiting customer-average waiting time
  in the system.

Reference S. Stidham, A last word on L ?W,
Operations Research, 1974.
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W Q / E

• In our case,
• Q is the limiting time-average number of admitted
  items in the system.
• E is the limiting time-average rate of arriving
  items that can be admitted.
• W is the limiting mean waiting time of an
  arbitrary admitted item in the system.

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Numerical Results Mean Waiting Time (in T)
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CPU Time Mean Waiting Time
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Milestone

• Finite buffer TDMA model with state dependent
  service
• Exact waiting time distribution of an arbitrary
  admitted item in the system
• Mean waiting time of an arbitrary admitted item
  in the system, using Little formula
• Future work

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Future Work

• Variable message length (Multi-packet message),
  instead of constant message length (Single-packet
  message). Compound Poisson arrival process?
• Waiting time density in such a finite buffer TDMA
  system with constant service was obtained in
  Clare 1983. Need to check if it is also flawed.
  ?
• In situations where the assumption of Poisson is
  inappropriate. Markovian Arrival Process?

Thank you !