[PPT] - Stochastic Modeling and Analysis of Home Page Multichannel SW-ARQ PowerPoint Presentation

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Stochastic Modeling and Analysis of Multichannel SW-ARQ

Candidate: Jun Li, M.Sc. Supervisor: Dr. Yiqiang Q. Zhao School of Mathematics and Statistics Carleton University

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Outline

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1. Motivations and List of Contributions
2. Models of Multichannel SW-ARQ
3. Resequencing Analysis of MSW-ARQ-inS
4. Packet Delay Analysis
5. Conclusion and Future Work

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Wireless Ad-Hoc Data Networks

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Each node generates independent data
Each node can communicate with its neighboring nodes
Data transmitted in packets are error-intolerable
Unreliable wireless links (typically bi-directional) can cause

packets to be lost or erroneously received

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Automatic Repeat Request (ARQ)

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A technique of recovering the lost and the erroneously received

packets through retransmission of packets

Assumptions:
1. Bidirectional channels: forward and backward (feedback)
2. Reliable error detection scheme (e.g., cyclic redundancy

check (CRC) code)

Classical ARQ protocols:
1. Stop-and-wait ARQ (SW-ARQ)
2. Go-back-N ARQ (GBN-ARQ)
3. Selective-repeat ARQ (SR-ARQ)

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Studies on ARQ over a Single Channel

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A single channel connects two nodes (transmitter and receiver)
One ARQ protocol, SW-ARQ or GBN-ARQ or SR-ARQ, is

used for packet error correction

Packet errors in the channel may either occur independently or

be correlated over time

ARQ protocols over a single channel with both time-invariant

(e.g., [Anagnostou, Fantacci, Rosberg]) and time-correlated (e.g., [Badia, Kim, Vuyst]) error models have been extensively studied

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Studies on ARQ over Multiple Channels

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An arbitrary number of parallel channels connect the transmitter-

receiver pair (e.g., a multiple-input multiple-output (MIMO) system) to increase data transmission rate

Multichannel ARQ protocols have been proposed to use in real-

world communication systems

Assumption: error models of parallel channels are independent
f each other
ARQ protocols for the time-invariant error model of channels

have been studied in [Chang, Ding, Fujii, Shacham, Wu], but far less completely

No studies on ARQ over time-varying multiple channels have

been reported

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List of Contributions

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Stochastic modeling of multichannel SW-ARQ
Derivation of the probability generating function for the rese-

quencing buffer occupancy

Derivation of the probability mass function of the resequencing

delay

Development of a systematic method for analysis of the packet

delay

Identification of the impact on the model performance numeri-

cally and through simulation

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Outline

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1. Motivations and List of Contributions
2. Models of Multichannel SW-ARQ
3. Resequencing Analysis of MSW-ARQ-inS
4. Packet Delay Analysis
5. Conclusion and Future Work

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MSW-ARQ Model

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channel M channel 2 channel 1

feedback channel

TX RX

channel 1

1

channel 2

2 5

channel 3

3 6 4* 4 5* 6 7 6* 5 6 8 7

Integer numbers in boxes are sequence numbers of packets. *, TX, RX denote transmission error,transmitter, receiver, respectively. (M = 3, m = 5)

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Model Description of MSW-ARQ

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1. The transmitter is saturated with packets ordered by their se-

quence numbers

2. Transmission rate for each channel is 1 packet per slot
3. Transmitter sends out M packets every m slots, one packet per

(forward) channel

4. Packet transmitted over a channel could be erroneously received

according to some channel error model (discussed next)

5. Acknowledgement packets (ACKs/NACKs) are sent via the

error-free feedback channel from receiver to transmitter

6. Transmitter retransmits the packet with the smallest sequence

number among all erroneously received packets as well as pack- ets with sequence numbers larger than that

7. Receiver only delivers a correctly received packet for which all

packets ahead of it were received without an error to the upper layer

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MSW-ARQ-inS Model

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TX RX

channel 1

1

channel 2

2 5*

channel 3

3 6 4* 4* 5 7* 7 8* 4 10 9 8

Integer numbers in boxes are sequence numbers of packets. *, TX, RX denote transmission error,transmitter, receiver, respectively. time

tm (t+1)m

Error detected and negligible time for sending NACKs

(t+2)m (t+3)m (t+4)m 6 6 5

contents of the resequencing buffer

1. – 5. Same as MSW-ARQ
6. Only erroneously received packets are retransmitted
7. The resequencing buffer is provided at the receiver for buffering undeliver-

able packets

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Channel Models & Packet Scheduling

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Channel Model: defines the statistical property of transmission

errors for a channel when transmitting packets

1. iid model: A time-invariant packet error rate is assumed for a

channel

2. Markov model: Error rate is time-varying but determined by

a two-state Markov chain

Packet Scheduling Policy: is a packet-to-channel assignment

rule when transmitter simultaneously transmits M packets over M channels

1. Static scheduling policy: An old packet is retransmitted over

the same channel as originally assigned one

2. Dynamic scheduling policy: The packet with the kth smallest

sequence number is assigned to the channel having the kth smallest error rate Note: Our analytical results are based the dynamic scheduling pol-

icy. Simulation results for the static scheduling policy are given for

performance comparisons.

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Outline

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1. Motivations and List of Contributions
2. Models of Multichannel SW-ARQ
3. Resequencing Analysis of MSW-ARQ-inS
4. Packet Delay Analysis
5. Conclusion and Future Work

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Resequencing for iid Channel Model

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We assume the error rate of channel i is pi ∈ (0, 1); pi might be

different from pj for i = j; the M channels are numbered from 1 to M such that p1 ≤ p2 ≤ · · · ≤ pM

We define,

P =     1 · · · 0 P10 P11 · · · 0 P20 P21 P22 · · · 0 . . . . . . . . . ... . . . PM,0 PM,1 PM,2 · · · PM,M     , (1) and Q = 1 0 q T

=

    1 · · · 0 q1 p1 · · · 0 q2 Q21 Q22 · · · 0 . . . . . . . . . ... . . . qM QM1 QM2 · · · QMM     . (2)

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Resequencing for iid Channel Model (Cont.)

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Let qk = 1 − pk, Ωi = {1, · · · , i}, Bi,j is a subset of Ωi of size j,

Bc

i,j = Ωi \ Bi,j, k∈Φ qk = k∈Φ pk = 1.

In (1),

Pij =

Bi,j⊆Ωi

 

k∈Bi,j

pk

k∈Bc

i,j

qk   . (3)

In (2),

Qij = pi

Bi−1,j−1⊆Ωi−1

 

k∈Bi−1,j−1

pk

k∈Bc

i−1,j−1

qk   . (4)

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Resequencing for iid Channel Model (Cont.)

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The probability generating function GBr(z) of the resequencing buffer oc-

cupancy Br is given in the following theorem Theorem 1 GBr(z) = PM0 +

M

i=1

∞

n=1

i−1

x1=0

x1

x2=0

· · ·

xn−2

xn−1=0

 

M

j=1

(pj + qjz)µj   P[C(n, i, x1, · · · , xn−1)], (5) where, P[C(n, i, x1, · · · , xn−1)] =

i−1

δ=0

PMδ

Pi−1,x1Px1,x2 · · · Pxn−1,0pi

n−1

k=1

pxk+1

,

(6) and µj = n − 1 − max{k : 0 ≤ k ≤ n − 1, xk + 1 ≥ j}, if j ≤ i, n,

therwise.

(7)

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Resequencing for iid Channel Model (Cont.)

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The probability mass function of the resequencing delay Dr is given in the

following theorem Theorem 2 P [Dr = xm] = M

i=1 P[Λi] i k=1 Pk−1,0qkSik,

if x = 0, M

i=2 P[Λi] i k=2 qkSik

k−1

j=1 P (x) k−1,jPj0

,

if x ≥ 1, (8) where, P[Λi] = 1 1 − PMM

M

δ=(M+1)−i

PM,M−δ δ , (9) and S ≡

∞

n=0

Tn = (I − T)−1. (10)

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Resequencing for Markov Channel Model

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We assume the error rate of each channel follows a Markov chain

with the state space {eG, eB} (0 < eG < eB < 1) and the transi- tion probability matrix

α

1 − α 1 − β β

.
Let i = {(i, 0), (i, 1), · · · , (i, M)},

x

y

be the binomial coeffi-

cient of x choosing y, ¯ h = M −h, ¯ eG = 1−eG, ¯ eB = 1−eB, ¯ α = 1 − α, ¯ β = 1 − β

We define

P =     P0,0 O O · · · O P1,0 P1,1 O · · · O P2,0 P2,1 P2,2 · · · O . . . . . . . . . ... O PM−1,0 PM−1,1 PM−1,2 · · · PM−1,M−1     (11) where, O represents the zero matrix,

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Resequencing for Markov Channel Model (Cont.)

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P(0,h),(0,k) =

h

l=0

h l

βl ¯

βh−l ¯ h k − l

¯

αk−lα

¯ h+l−k

(12)

and, P(i,h),(j,k) = f(i, j, h)P(0,h),(0,k) (13) for which f(i, j, h) =      i

j

ej

G¯

ei−j

G ,

if i ≤ ¯ h,

j

l=0

¯

h l

el

G¯

e

¯ h−l G

i−¯

h j−l

ej−l

B ¯

ei−¯

h+l−j B

, else, (14) for 1 ≤ i ≤ M − 1, 0 ≤ j ≤ i and 0 ≤ h, k ≤ M.

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Resequencing for Markov Channel Model (Cont.)

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The probability generating function GBr(z) of the resequencing buffer oc-

cupancy Br is given in the following theorem Theorem 3 GBr(z) = P[A0] +

∞

n=1
(x0,y0)∈Θ

· · ·

(xn−1,yn−1)∈Θ

M

yn=0

n−1

l=0

GB(l)

r,n(z)

P[C(n, x0, y0, · · · , xn−1, yn−1, 0, yn)],

(15) where, P[A0] = M

y=0 f(M, 0, y)πy, Θ = {0, 1, · · · , M − 1},

GB(l)

r,n(z) =

(eB + z¯ eB))M−1−xl, if xl + yl ≥ M − 1, (eB + z¯ eB)yl(eG + z¯ eG)M−1−xl−yl, else, (16) P[C(n, x0, y0, · · · , xn−1, yn−1, 0, yn)] = n−1

l=0

δl  

M

y=0

πy

x0

x=0

P(M,y),(x,y0)   P(x0,y0),(x1,y1) · · · P(xn−1,yn−1),(0,yn)(17) in which, δl = eB, if xl + yl > M − 1, eG, else.

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Numerical and Simulation Results

– Mean Resequencing Buffer Occupancy for Time-invariant Channels

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2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 18 20 M Mean resequencing buffer occupanc y Dynamic rule Static rule

(Note: M is the number of parallel channels)

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Numerical and Simulation Results (Cont.)

– Mean Resequencing Delay for Time-invariant Channels

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0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.2m p Mean resequencing delay M=4 M=8 M=16 0.4m 0.6m 1.4m 1.2m m 0.8m 1.8m 1.6m

(Note: p = 1

M

i=1 pi)

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Numerical and Simulation Results (Cont.)

– Mean Resequencing Buffer Occupancy for Time-Varying Channels

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1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 K Mean resequencing buffer occupancy Markov with dynamic rule iid Markov with static rule

(Note: K = eB

eG)

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Summary on Resequencing Analysis of MSW-ARQ-inS

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The mean resequencing buffer occupancy for both time-varying and time-

invariant channels benefits from the dynamic scheduling

The mean resequencing buffer occupancy and the mean resequencing delay

for both channel models grow with the increase of either the number of channels or the average error rate of channels

When the time-invariant channel error rates are assumed, the mean re-

sequencing buffer occupancy and the mean resequencing delay decrease as the error rates of different channels become more different.When the Markov model is assumed, they decrease as the absolute difference be- tween the two error states becomes larger

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Outline

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1. Motivations and List of Contributions
2. Models of Multichannel SW-ARQ
3. Resequencing Analysis of MSW-ARQ-inS
4. Packet Delay Analysis
5. Conclusion and Future Work

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Packet Delay of MSW-ARQ for iid Model

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We define the following matrix

P =              1 · · · · · · 0 q1 p1 · · · · · · 0 q1q2 p2q1 p1 · · · · · · 0 . . . . . . . . . ... . . . ... . . .

i

k=1

qk pi

i−1

k=1

qk pi−1

i−2

k=1

qk · · · p1 · · · 0 . . . . . . . . . ... . . . ... . . .

M

k=1

qk pM

M−1

k=1

qk pM−1

M−2

k=1

qk · · · pM−i+1

M−i

k=1

qk · · · p1              . (18)

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Packet Delay of MSW-ARQ for iid Model (Cont.)

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The distribution function of the packet end-to-end delay D is given by the

following theorem Theorem 4 The steady-state probability distribution of the packet delay D for an arbitrary packet is given by d(x) ≡ P

D = m + 1

2 + mx

=

          

1 1−PMM M

i=1
Pi0

M

δ=(M+1)−i

PM,M−δ δ

,

if x = 0,

1 1−PMM M

i=1
i
j=1

P[τj = x]Pij

M

δ=(M+1)−i

PM,M−δ δ

,

if x ≥ 1, (19) where, P[τi = t] = Pi0, if t = 1, i

k=1 P[τk = t − 1]Pik,

if t ≥ 2. (20)

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Packet Delay of MSW-ARQ-inS for iid Model

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The packet end-to-end delay ¯

D of MSW-ARQ-inS is given by the following lemma Lemma 1 The steady-state probability distribution function of ¯ D has the same expression as (19) and (20), where Pij is given by (3).

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Numerical and Simulation Results

—- Average Packet Delay vs. M ————————————————————————————————–

2 4 6 8 10 12 14 16 5 10 15 20 25 30 M Average packet delay MSWARQinS with Dynami c MSWARQinS with Static MSWARQ with Dynami c MSWARQ with Static

(Note: M is the number of parallel channels)

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Numerical and Simulation Results (Cont.)

—- Average Packet Delay vs. p ————————————————————————————————–

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 5 10 15 20 25 30 p Average packet delay MSWARQinS with Dynami c MSWARQinS with Static MSWARQ with Dynami c MSWARQ with Static

(Note: p = 1

M

i=1 pi)

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Numerical and Simulation Results (Cont.)

—- Average Packet Delay vs. ∆ ————————————————————————————————–

1.1 1.2 1.3 1.4 1.5 1.6 1.7 5 10 15 20 25 Average packet delay MSWARQinS with Dynami c MSWARQinS with Static MSWARQ with Dynami c MSWARQ with Static

(Note: ∆i = pi+1

pi , i = 1, · · · , M − 1 and we assume ∆ = ∆1 = · · · = ∆M−1)

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Summary on Packet Delay Analysis

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1. For both MSW-ARQ and MSW-ARQ-inS, the dynamic schedul-

ing always outperforms the static scheduling in terms of the mean packet delay performance

2. With the dynamic scheduling, MSW-ARQ-inS always outper-

forms MSW-ARQ

3. When the dynamic scheduling is applied, the mean packet de-

lay of both models increases with the average error rate p but decreases as the variance in the error rates increases, and the number of parallel channels has only an insignificant impact on the mean packet delay

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Conclusion

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1. We developed two stochastic systems, which are

adaptive to real-life implementations and mathemat- ically tractable, for modeling SW-ARQ over multi- ple parallel channels

2. For multiple channels of time-invariant error rates,

we conducted resequencing analysis for MSW- ARQ-inS and packet end-to-end delay analysis for both systems

3. For multiple channels of time-varying error rates,

we derived the probability generating function of the resequencing buffer occupancy of MSW-ARQ- inS

4. We investigated the impact of system parameters on

the mean value performance and conducted perfor- mance comparisons of the two packet transmission scheduling policies numerically, and through simu- lation

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Future Work

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1. For multiple channels of time-varying error rates,

the resequencing delay performance of MSW-ARQ- inS and the end-to-end delay performance of both systems will be evaluated

2. Performance evaluation of SR-ARQ over multiple

channels of time-varying channel error rates

3. Performance study on ARQ protocols over multiple

channels with both different transmission rates and different error rates

4. The models studied in this thesis can be general-

ized to allow the transmitter to send more than one copy of a packet either simultaneously over multiple channels or when channels are idle. We will conduct performance comparisons between the generalized models and the original ones

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References ————————————————————————————————–

Anagnostou M.E. Anagnostou, and E.N. Protonotarios, “Performance Analysis of the Selective Repeat ARQ Protocol,” IEEE Trans- actions on Communications, vol. 34, no. 2, pp. 127–135, 1986. Badia L. Badia, M. Rossi, and M. Zorzi, “SR ARQ Packet Delay Statistics on Markov Channels in the Presence of Variable Arrival Rate,” IEEE Transactions on Wireless Communications, vol. 5, no. 7, pp. 1639–1644, 2006. Chang J.-F. Chang, and T.-H. Yang, “Multichannel ARQ Protocols,” IEEE Transactions on Communications, vol. 41, no. 4, pp. 592–598, 1993. Ding Z. Ding, and M. Rice, “ARQ Error Control for Parallel Multichannel Communications,” IEEE Transactions on Wireless Commu- nications, vol. 5, no. 11, pp. 3039–3044, 2006. Fantacci R. Fantacci, “Mean Packet Delay Analysis for the Selective Repeat Automatic Repeat Request Protocol with Correlated Arrivals and Deterministic and Nondeterministic Acknowledgement,” Telecommunication Systems, no. 9, pp. 41–57, 1998. Fujii S. Fujii, Y. Hayashida, and M. Komatu, “Exact Analysis of Delay Performance of G0-Back-N ARQ Scheme over Multiple Parallel Channels,” Electronics and Communications in Japan, part 1, vol. 84, no. 9, pp. 27–41, 2001. Kim J.G. Kim, and M.K. Krunz, “Delay Analysis of Selective Repeat ARQ for a Morkovian Source over a Wireless Channel,” IEEE Transactions on Vehicular Technology, vol. 49, no. 5, pp. 1968–1981, 2000. Rosberg Z. Rosberg, and N. Shacham, “Resequencing Delay and Buffer Occupancy Under the Selective-Repeat ARQ,” IEEE Trans- actions on Information Theory, vol. 35, no. 1, pp. 166–173, 1989. Shacham N. Shacham, and B.C. Shin, “A Selective-Repeat-ARQ Protocol for Parallel Channels and Its Resequencing Analysis,” IEEE Transactions on Communications, vol. 40, no. 4, pp. 773–782, 1992. Vuyst S.D. Vuyst, “Analysis of Discrete-Time Queueing Systems with Multidimensional State Space,” Ph.D. thesis, Gent University, 2006. Wu W.-C. Wu, S. Vassiliadis, and T.-Y. Chung, “Performance Analysis of Multi-Channel ARQ Protocols,” in Proceedings of the 36th Midwest Symposium on Circuits and Systems, vol. 2, pp. 1328–1331, 1993.

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