Analysis of Latency for Reliable End-to-End Batch Transmission in - PowerPoint PPT Presentation

Analysis of Latency for Reliable End-to-End Batch Transmission in Multi-Rate Multi-Hop Wireless Networks Teeraw at I ssariyakul ( teeraw at@trlabs.ca) Ekram Hossain ( ekram @ee.um anitoba.ca) Attahiru Sule Alfa ( alfa@ee.um anitoba.ca)

Outline • Background and Motivations • Modeling end-to-end transmission (Main contribution) • Numerical results • Summary, conclusions, and future studies

Why Multi-Hop? • Use short range communications – Increase data rate – Reduce delay Base st at ion – Reduce energy consumption • Multi-hop relay data from the base station to the mobile – Increase coverage of Short Range service area – Better load balance Long Range

What About Ad Hoc ? • Distributed medium access control protocols (e.g., IEEE 802.11) lead to degradation in an end-to-end level • More hops = > Geometric increase in end-to-end error • Limited energy consumption

Wireless Channel and ARQ • Random Error in Data Transmission • Adaptive Modulation and Coding – Bound average hop-level packet error probability – High SNR: increase transmission rate – Low SNR: decrease transmission rate • Automatic Repeat reQuest (ARQ) with I NFINITE persistence One successf ul 5 t r ansmissions end-t o-end deliver y SOURCE DESTI NATI ON

Main Contribution • Chain topology with 3 nodes N packets • Both hops can transmit at the same time (e.g., ODMA) • Objective: . . . – Note: Different application may need different delay requirement – Find distribution of the latency to delivery all N packets to the destination. Source Destination

Main Contribution (cont.) • Implication of delay distribution – E.G., the batch becomes useless after k time slots, – Pr{ delay < = k} = Probability to deliver the batch before it becomes useless.

Absorbing Markov Process TRANSI ENT STATES 1 ABSORBI NG STATE p AC X1 p AB ... ... A B 1 p BA p CA Xn p ij is t he t r ansit ion probabilit y f r om st at e i t o j • Start points: any state • Finish points: any absorbing state

Absorbing Markov Process • Transition probability matrix ( P ) To A B … X From A p AA p AB … p AX Q R B p BA p BB … p BX = P = 0 I . . . . . . . . . I X 0 0 … ( α,α 0 ) = the initial probability matrix

Phase (PH) Type Distribution • PH distribution: distribution of time to absorption in an absorbing Markov process • Let k be the number of transitions to reach the absorbing state. PMF CDF α =  α =  ; k 0 ; k 0 = = 0 0   F f − α + − ⋅ ⋅ > k α ⋅ − ⋅ > k 1 k  α Q e k 1 1 ; k 0  Q R ; k 0 0 Expectation − 1 = ⋅ − ⋅ α E[k] ( I Q ) 1

System Model (details) • Wireless channel – Independent and identically distributed (i.i.d.) – Channel state m, randomly chosen from [ 1,… ,M] – State = m: transmit m packets – Each transmitted packet is in error with probability p err • Use ARQ with infinite persistence at each node.

Absorbing Markov Model • Model the transmission process as an absorbing Markov Chain Multi-Hop Network Markov Chain Starting point N packets are Initial state supplied to the source node Finishing point All N packets reach Absorbing the destination node state • Latency = Time to absorption – PMF, CMF, and Expectation

Queuing Model 1 2 ... 3 • Absorbing Markov chain (X1,X2) • Xi = buffer size of node i Multi-Hop Network Markov Chain Starting point N packets are supplied Initial state = to the source node (N,0) Finishing point All N packets reaches Absorbing the destination node state = (0,0)

Mathematical Model The state where N packets are • Final steps supplied to the source node 1. Find Relevant Matrices • Initial probability matrix: α = e i = [ 0 … 0 1 0 … 0] • Transition probability matrix: P (next page) 2. Use the formulae for absorbing Markov process to find • PMF • CMF • Expectation

Statistics for Latency • Latency = Time to absorption PMF CDF α =  α =  ; k 0 ; k 0 = = 0 0   F f − α + − ⋅ ⋅ > k α ⋅ − ⋅ > k 1 k  α Q e k 1 1 ; k 0  Q R ; k 0 0 Expectation − 1 = ⋅ − ⋅ α E[k] ( I Q ) 1

Statistics for Latency • Latency = Time to absorption

Transition Probability Matrix (P) I 0 From (X1,X2) P = To (X1,X2) R Q 00 01 02 03 10 11 12 20 21 30 00 1 absorbing state 01 X1 does not change R Q 02 B A '00 03 X1 decreases 10 A 10 A 11 11 12 20 A 20 A 21 A 22 21 30 A 30 A 31 A 32 A 33 initial state

Transition Probability Matrix 00 01 02 03 10 11 12 20 21 30 00 X1 = 0, X2 = 2 01 02 03 10 11 12 X1 = 3, 20 X2 = 0 21 30 p i (s i ) = probability that node i successfully transmit s packets

Transition Probability Matrix 00 01 02 03 10 11 12 20 21 30 00 01 02 03 10 11 12 20 21 30 p i (s i ) = probability that node i successfully transmit s packets

Successful Transmission Probability S i > X i S i <= X i Complementary cumulative distribution function Probability that s i packets are successfully transmitted, given that total transmitted packet is m

Model Validation E[k] M (Max Tx Rate) = N (Batch Size) Equally Likely Channel Batch Size (N)

Effect of Number of Channel States Deep slope: High improvement Slope decreases: E[k] Xi< m, for some time slot N = 10 Max. Tx. Rate (M)

Cumulative Distribution Function 95% CDF (F k ) End-to-end latency (k)

Minimum Latency Latency for 95% Delivery Batch Size (N)

Summary and Conclusions • End-to-end latency distribution in a multi-hop wireless network as a function of – link-error probability, – transmission rate, and – end-to-end latency distribution • Validate using simulation • Expected latency does not guarantee high probability of batch delivery • Increasing max. Tx rate or channel states – Decreasing Latency – Rate under is underutilized, since packets in the buffer < current transmission rate – Decreasing rate of decrease in latency

Further Studies • Other ARQ policies (e.g., limited persistence) see WN27-3 • More realistic channel model (e.g., Rayleigh Fading or FSMC) • Channel Access Policies • Extension to window-based congestion control (window= batch) • Steady State Analysis

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