Latency-Reliability Tradeoff for Different Hop-Level ARQ-based Error Recovery in a Multi-Hop Wireless Network Teeraw at I ssariyakul ( teeraw at@trlabs.ca) Ekram Hossain ( ekram @ee.um anitoba.ca) Attahiru Sule Alfa ( alfa@ee.um anitoba.ca)
Outline • Introduction • System Model and Main Contribution • Modeling end-to-end transmission • 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
Previous Work • TCP throughput analysis under collision free and error-free I EEE 802.11 two-hop (3 nodes) network (WCNC’04) • Model for number of transmissions in an H-hop chain topology for a single packet (Globecom’04) • A similar model with rate adaptation and infinite persistent ARQ (WN27-1)
System Model • Batch transmission with N packets • Chain topology with 3 nodes N packets • Both hops can transmit at the same time (e.g., ODMA) . . . • Packet error probability is i.i.d. with probability p err • Different types of ARQ at each node Source Destination
Main Contribution • Delay analysis considering – Queuing, – non-zero error probability, and – ARQ • Send N packets • s packets are delivered successfully (s= { 0,1,… ,N} ) • N-s packets are not delivered • Objective – Find Pr{ s packets are delivered} – Find pmf (probability mass function) of associated delay
Hop-Level ARQ • Possible ARQ policies: – ARQ 0 : zero retransmission (stop immediately) – ARQ ∞ : infinite retransmission (never stop) – ARQ F : finite retransmission (stop after M failures) – ARQ P : probabilistic retransmission with infinite persistence (stop with probability d after each failure) • In this paper, we use only ARQ 0 , ARQ ∞ , ARQ P
Hop-Level ARQ • If transmission stops, – The transmitting node will reset itself. – It will flush the buffer, and will not receive any incoming packet. – Transmission at the other nodes can still continue. • Source stops: the process continues • Intermediate node stops: The process ends Source Destination
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 Delay PMF α = ; k 0 = 0 f − > k α Q k 1 R ; k 0 Absorbing Probability Expected Delay ( ) R − = − 1 α = − − f I Q α 2 E [k] ( I Q ) R
Queuing Model 1 2 ... 3 • Absorbing Markov chain (X1,X2,X3) • Xi = buffer size of node i Multi-Hop Network Markov Chain Starting point N packets are supplied Initial state = to the source node (N,0,0) Finishing point No packet in the Absorbing network (X1= 0,X2= 0) state = (0,0,s)
Mathematical Model The state where N packets are • Final two 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
Infinite Retransmission ARQ ( ARQ ∞ ) • Batch size N • State Space S N = { (X1,X2,X3): X1+ X2= N, X3= N-X1-X2} • TPM – Packets in the system always decrease – Lower-triangular – will later be used to derive ARQ P
Finding Main Statistics • PH-Distribution ( ) R − = − 1 α f I Q α = ; k 0 = 0 f − > k α Q k 1 R ; k 0 = − − α 2 E [k] ( I Q ) R • Main Statistics ( ) = f d e f D d – Delay PMF: ( ) = f ( m 1 , ) f M m – Pr{ m pkts successfully TX} : [ ] e = – Expected Latency: E [D] E d
Transition Probability Matrix for ARQ ∞ I 0 • For ARQ ∞ : P = R Q (X1,X2,X3) absorbing state X1 does not change R Q X1 decreases initial state
Transition Probability Matrix for ARQ ∞ S = Success, F = Fail
Transition Probability Matrix for ARQ ∞ S = Success, F = Fail
Probabilistic Retransmission ARQ ( ARQ P ) • Start with N packets in the system (S N ) • If k packets are dropped, S N -> S N-k • State Space: S 1 U S 2 U … U S N • TPM is lower-triangular
Probabilistic Retransmission ARQ ( ARQ P ) Stay in S N = TPM of ARQ ∞ All packets in the system S i are delivered Drop 1 packet Drop 2 packets
Finding Main Statistics • PH-Distribution ( ) ω − = − Ω 1 α f I α = ; k 0 = 0 f Ω − ω > k α k 1 ; k 0 = − Ω − ω α 2 E [k] ( I ) • Main Statistics ( ) = f d e f D d – Delay PMF: ( ) = f ( m 1 , ) f M m – Pr{ m pkts successfully TX} : [ ] e = – Expected Latency: E [D] E d
Probabilistic Retransmission ARQ ( ARQ P ) • RF = Node1 reset and Node2 fail • RS = Node1 reset and Node2 success • R1 = Node1 reset, R2 = Node2 reset • S1 = Node1 success, S2 = Node2 success
Probabilistic Retransmission ARQ ( ARQ P ) • RF = Node1 reset and Node2 fail • RS = Node1 reset and Node2 success • R1 = Node1 reset, R2 = Node2 reset • S1 = Node1 success, S2 = Node2 success Q 1 Q 2 Q 3
Numerical Results Several packets might be dropped during one connection reset E[M] and E[D]
Numerical Results E[M]/E[D] Decrease in slope
Numerical Results E[D]
Numerical Results 95% ( ) = [ ] 52 . 12 % F D E D CDF (F k ) End-to-end latency (k)
Numerical Results p=0.7 PMF (f M (m)) p=0.9
Summary • End-to-end latency distribution in a multi-hop wireless network in terms of – link-error probability, – hop-level ARQ parameters, and – end-to-end latency distribution • Validate using simulation • Retransmission – Increases reliability – Increases end-to-end delay • Tradeoff is quantified by the proposed model • Expected latency does not guarantee high batch delivery
Conclusions • Retransmission – Increases reliability – Increases end-to-end delay • Tradeoff is quantified by the proposed model • Expected latency does not guarantee high batch delivery • High batch delivery can be obtained at the expense of increasing latency
Further Studies • Multi-rate transmission (WN27-1) • 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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