A Network Calculus for Multi-Hop Fading Channels Hussein Al-Zubaidy - PowerPoint PPT Presentation

A Network Calculus for Multi-Hop Fading Channels Hussein Al-Zubaidy J¨ org Liebeherr Almut Burchard University of Toronto

Performance Analysis of Multihop Wireless Network z Cross traffic Cross traffic Cross traffic Through traffic n N 1 . . . . . . Intermediate nodes are store and Cross traffic forward relays Through A fading channel is characterised traffic by its channel capacity Node model Fading channel capacity

Fading Channel Capacity Channel capacity [Shannon 1948] C ( γ ) = W log(1 + γ ) γ | h | 2 for fading channels γ = ¯ Channel gain h is a complex r.v. Q: How do fading channel properties affect multihop network performance?

Network Model Cross traffic Cross traffic Cross traffic j j A D . . . n Fluid-flow traffic, discrete time y x h Arrival and service are independent z I.i.d. cross traffic at each node n N Time-varying random service that is equal to the Instantaneous channel capacity γ | h t | 2 � � C ( γ t ) = W log g ( γ t ) , γ t = ¯ Computing this service distribution is hard!

Related Work: Multihop network performance analysis Simplified channel models FSMC model [Wang and Moayeri 1995][Sadeghi et al 2008] more than two states models may not be tractable not easily extended to multihop networks ON-OFF model tractable but very simplified model used in queuing theory [Ishizaki 2007], network calculus [Ciucu 2011], effective bandwidth [Hasan,Krunz,Matta 2004] Effective capacity [Wu and Negi 2003] log-MGF of the channel capacity tractable only for low SNR where log(1 + γ ) ≃ γ Physical layer models [Hasna and Alouini 2003] outage probability for AF wireless relay network expression for MGF of end-to-end SNR not suitable for network analysis

Network Calculus D A S (min , +) dioid algebra Backlog: B ( s ) = A (0 , s ) − D (0 , s ) Delay: W ( s ) = inf { u ≥ 0 : A (0 , s ) ≤ D (0 , s + u ) } Dynamic server [Chang 2000] A S D A (0 , t ) D (0 , t ) ≥ inf u ≤ t { A (0 , u ) + S ( u, t ) } delay = W ( s ) D (0 , t ) = A ∗ S (0 , t ) backlog = B ( s ) s Network service: t S net ( τ, t ) = S 1 ∗ S 2 ∗ · · · ∗ S N ( τ, t )

Network Analysis in Bit Domain ... A ( t ) S 1 S N D ( t ) Bit domain Arrivals and departures are measured in bits X For fading channels, service is given in terms of log( g ( γ t )) Distribution of S is not easy to work with

SNR Domain Transfer domain ... S 1 S N (‘SNR domain’) ... Bit domain A ( t ) S 1 S N D ( t ) Service in terms of g ( γ t ) rather than log( g ( γ t )) – more X tractable SNR service S ( τ, t ) = � t − 1 i = τ g ( γ i ) resides in the SNR domain

SNR Domain Transfer domain ... A ( t ) S 1 S N D ( t ) (‘SNR domain’) ... Bit domain A ( t ) S 1 S N D ( t ) Service in terms of g ( γ t ) rather than log( g ( γ t )) – more tractable SNR service S ( τ, t ) = � t − 1 i = τ g ( γ i ) resides in the SNR domain

Our Approach Transfer domain ... D ( t ) A ( t ) S 1 S N (‘SNR domain’) log( X ) e X ... Bit domain A ( t ) S 1 S N D ( t ) SNR domain is governed by (min , × ) dioid algebra Network SNR server S net ( τ, t ) = S 1 ⊗ S 2 ⊗ · · · ⊗ S N ( τ, t )

(min , × ) Network Calculus Service: S ( τ, t ) = � t − 1 i = τ g ( γ i ) S A D Arrival: A ( τ, t ) = � t − 1 i = τ e a i � � Departure: D (0 , t ) ≥A⊗S ( τ, t )=inf τ ≤ u ≤ t A ( τ, u ) ·S ( u, t ) � � A (0 ,t ) Backlog: B ( t ) = log D (0 ,t ) Delay: W ( t ) = inf { u ≥ 0 : A (0 , t ) ≤ D (0 , t + u ) }

Computation of S 1 ⊗ S 2 Mellin transform: M X ( s ) = E [ X s − 1 ] For two independent servers t � M S 1 ⊗S 2 ( s, τ, t ) ≤ M S 1 ( s, τ, u ) · M S 2 ( s, u, t ) u = τ For N i.i.d. fading channels � N − 1 + t − τ � � t − τ , � M S net ( s, τ, t ) ≤ · M g ( γ ) ( s ) ∀ s < 1 t − τ Moment bound: Pr ( X ≥ a ) ≤ a − s M X (1 + s ), ∀ a, s > 0

Main Result: Statistical Performance Bounds Define min( τ,t ) � M ( s, τ, t ) = M A (1 + s, u, t ) · M S (1 − s, u, τ ) u =0 B ( t ) > b ε � � Backlog: Pr ≤ ε , where � 1 b ε = inf �� � log M ( s, t, t ) − log ε s s> 0 W ( t ) > w ε � � Delay: Pr ≤ ε , where � � M ( s, t + w ε , t ) inf ≤ ε s> 0

Cascade of N i.i.d. Rayleigh Channels Service for Rayleigh channels γ | h | 2 g ( γ ) = 1 + γ = 1 + ¯ | h | ∼ Rayleigh r.v. For i.i.d. Rayleigh fading channel � t − τ � e 1 / ¯ γ ¯ γ s − 1 Γ( s, ¯ γ − 1 ) M S ( s, τ, t ) = Arrivals: ( σ ( s ) , ρ ( s )) bounded arrivals [Chang 2000] M A ( s, τ, t ) ≤ e ( s − 1) · ( ρ ( s − 1) · ( t − τ )+ σ ( s − 1)) , s > 1 This traffic class includes Markov-modulated processes, effective bandwidth, etc.

Performance Bounds of N Rayleigh Channels Define: γ − s Γ(1 − s, 1 △ = e sρ ( s ) e 1 / ¯ γ ¯ V ( s ) γ ) ¯ � B ( t ) > b ε � Backlog: Pr ≤ ε , where net � �� σ ( s ) − 1 b ε � net = inf N log(1 − V ( s )) + log ε s s> 0 W ( t ) > w ε � � ≤ ε , where Delay: Pr � � e s ( − ρ ( s ) w ε + σ ( s )) 1 , ( V ( s )) w ε ( w ε ) N − 1 � � inf · min ≤ ε (1 − V ( s )) N s> 0

Numerical Results for N Rayleigh Channels Model parameters ∆ t = 1 ms W = 20 kHz ( σ, ρ ) bounded traffic σ = 50 kb ρ = 0 to 60 kbps ¯ γ = 0 to 40 dB N = 1 to 100 We used deterministically bounded traffic, hence, the only source of randomness is the fading channel!

Backlog Bounds for N Rayleigh Channels & . .<1*.1<7* .#!.1<7*A %'" :17 ! ;< ! *17.=,/>2<-.=<?17.6@=9 .$!.1<7*A .%!.1<7*A % ."!.1<7*A .#!!.1<7*A b ε $'" net vs. ¯ γ $ ρ = 30 kbps #'" ε = 10 − 4 # !'" ! . ! " #! #" $! $" %! %" &! ()*+,-*./0,11*2.345.6789 $" - --97/-798/ --!"-798/4 678 ! .9 ! /78-2+:1,9;-29<78-0=25 --$"-798/4 --%"-798/4 !# --#"-798/4 b ε !""-798/4 net vs. ρ !" γ = 10 dB ¯ ε = 10 − 4 # " - !" !# $" $# %" %# &" &# #" ## '(()*+,-(+./-012345

Backlog and Delays ! #! - - ! -A-$!-B=<;C-:0)-0:6) - ! -A-%!-B=<;C-:0)-0:6) - ! -A-$!-B=<;C-#!-0:6); ! # #! (i) ε ( b ) vs. ¯ γ - ! -A-%!-B=<;C-#!-0:6); - ! -A-$!-B=<;C-$!-0:6); 9:;;-<*:=+=>1>?@ - ! -A-%!-B=<;C-$!-0:6); ! $ buffer size = 400 kb #! Waterfall curves for loss ! % #! probability ! & #! - ! " #! #" $! $" %! '()*+,)-./+00)1-234-5678 ! #! . .A ! .B.#!.CD .A ! .B.$!.CD (ii) ε ( w ) vs. ¯ γ ! % #! .A ! .B.%!.CD .A ! .B.E!.CD :;<2,=;<1.>+<?,?;2;=@ N = 10 ! ' #! ρ = 20 kbps ! & #! Tighter delay bounds at higher SNR ! #$ #! ! #" #! . ! " #! #" $! $" %! %" ()*+,-*./0,11*2.345.6789

Conclusions New approach to analyze cascade of fading channels Analysis in SNR domain using (min , × ) dioid algebra Use Mellin transform and moment bound to compute end-to-end bounds Application to cascade of i.i.d. Rayleigh channels Explicit bounds in terms of the physical channel parameters Bounds scale linearly in N (min , × ) dioid algebra has potential applications in models with time varying channel models

Thank you Q & A

Delay bounds ! "! / / ! /=/&/+>?/-*./*-+. (iii) ε ( w ) vs. EtoE delay / ! /=/&/+>?/"!/*-+.7 / ! /=/"!/+>?/-*./*-+. – ρ = 20 kbps / ! /=/"!/+>?/"!/*-+.7 9:-01,:-*/;<-313:0:,2 ! $ / ! /=/"&/+>?/-*./*-+. "! / ! /=/"&/+>?/"!/*-+.7 – Effect of N on the / ! /=/#!/+>?/-*./*-+. / ! /=/#!/+>?/"!/*-+.7 violation prob. at low ! ' "! SNR is huge! ! ( "! / ! "! #! $! %! &! '! )*+ ! ,- ! .*+/+.012/3-4*+/5678 #!! / /=2+/2=8+ /#!/2=8+B /$!/2=8+B ;28 ! <= ! +28/8+3->/?=@28/7AB: (! /%!/2=8+B (iv) w ε net vs. ¯ /"!/2=8+B γ '! – ρ = 30 kbps – ε = 10 − 4 &! $! ! /

Fading Channels With Cross Traffic Leftover SNR service: A c ( t ) D c ( t ) S o ( τ, t ) = S ( τ, t ) A c ( τ, t ) S ( τ, t ) A o ( t ) D o ( t ) Dynamic SNR server: M S o ( s, τ, t ) = M S / A c ( s, τ, t ) = M S ( s, τ, t ) · M A c (2 − s, τ, t ) N-node: � N − 1 + t − τ � M S o, net ( s, τ, t ) ≤ e (1 − s ) · Nσ c (1 − s ) t − τ M g ( γ ) ( s ) e (1 − s ) · ρ c (1 − s ) � t − τ , � · s < 1

Bounds of Rayleigh Channels With Cross Traffic 1 End-to-end Backlog of the through flow � � �� σ o ( s ) + Nσ c ( s ) − 1 b ε � � o, net ( t ) ≤ inf N log 1 − V o ( s ) + log ε s s> 0 2 Delay bound, we estimate for w ǫ ≥ 0 � � e s ( − ρ o ( s ) w + σ o ( s )+ Nσ c ( s )) 1 , ( V o ( s )) w ǫ ( w ǫ ) N − 1 � � inf · min ≤ ǫ (1 − V o ( s )) N s> 0 where, V o ( s ) = e s · ( ρ o ( s )+ ρ c ( s ) e 1 / ¯ γ ¯ γ − s Γ(1 − s, ¯ γ − 1 )