Randomized Channel Access Reduces Network Local Delay Wenyi Zhang USTC Joint work with Yi Zhong (Ph.D. student) and Martin Haenggi (Notre Dame) 2013 Joint HK/TW Workshop on ITC CUHK, January 19, 2013
Acknowledgement K. C. Wong Education Foundation Conference Sponsorship Funding agencies: MST China, NSF China, MOE, Chinese Academy of Sciences
Outline Introduction (A Bit More on) Local Delay System Model Analysis Wrap-up
Introduction Local Delay: Time (# of time slots) until a unit amount of data (a packet) is successfully delivered over one hop In a random network, LD is a R.V., --- different communication pairs experience different LDs
Interference Correlation: Temporal correlation Correlation of interferences measured at a location (o) among different time slots How come? --- Common nodes may transmit in multiple time slots Introduction Spatial correlation Correlation of interferences measured at nearby locations (o and ) How come? --- Spatial deployment of nodes fixed once a network is set up
Aim of Study Understand the impact of interference correlation on local delay Find ways of reducing local delay through reducing interference correlation Channel access randomization is key!
First systematic framework: A Bit More on Local Delay F. Baccelli and B. Blaszczyszyn, Stochastic Geometry and Wireless Networks, Vol. II, Applications, Chap. 17, Foundations & Trends in Networking, NOW, 2009 F. Baccelli and B. Blaszczyszyn, A new phase transition for local delays in MANETs, IEEE INFOCOM 2010 Concrete case study on spatial Poisson networks with nearest-neighbor transmission rules: M. Haenggi, The local delay in Poisson networks, IEEE Trans. Inform. Theory, to appear
Wireless Contention Phase Transition In a nutshell, the mean local delay may be infinite for certain network parameters! Implication: LD as a R.V. has a heavy tail This phase transition phenomenon occurs for Static network and density of transmitters exceeding a threshold Strong temporal interference correlation In the infinite mean LD regime, the suffered receivers possess an exceedingly poor experience, even though most other receivers may enjoy high rates
An Example of Phase Transition Potential transmitters: PPP Φ of intensity λ Potential receivers: PPP Φ of intensityλ r r = qλ Each transmitter transmits with probability p =1 q Result [Haenggi12IT, Thm. 2]: The mean LD from a typical potential transmitter to its nearest potential receiver is for pq δ 2 < π / γ, and infinite otherwise. α > 2 δ = / α Path loss exponent is, and 2 < 1 So LD suffers when too many transmitters decide to turn on D 1 π p π γpq = δ 2
Randomized channel access Frequency-hopping ALOHA Correlation Shredder: Randomization time slot k time slot k + 1 Benefits Reduce interference correlation (key benefit) Reduce net amount of interference Frequency-hopping further reduces noise (minor benefit)
System Model Network Model Locations : Poisson bipolar model Physical layer: SINR threshold model Notations Φ = { i } α h k, x N 0 θ W x locations of transmitters path loss exponent fading coefficients between x and o in time slot k power spectral density of thermal noise SINR threshold for successful decoding total bandwidth Normalization A packet needs exactly one time slot to transmit if θ=1 (i.e., 0 db) and if the entire band of bandwidth W is used.
Frequency-hopping Frequency-hopping Total bandwidth W is divided into N equal sub-bands In each time slot a sub-band is uniformly randomly selected Number of successful time slots needed for transmitting a packet is N/log 2 (1+ θ) Interference and SINR Aggregated interference in time slot k is : Indices of sub-bands occupied by x in time slot k SINR in time slot k is
ALOHA ALOHA Transmitting probability p Entire band is used when transmitting Number of success time slots needed is 1/log 2 (1+ θ) Interference and SINR The interference in time slot k is : transmitting set in time slot k The SINR in time slot k is
Key idea Local Delay Analysis Transmission success events among time slots are conditionally independent given and the conditional success probability is Conditioned upon, the total number of time slots elapsed until a success event occurring is geometric with parameter Mean local delay is obtained by de-conditioning on Frequency-hopping : Palm expectation Mean of geometric R.V. ALOHA :
Taking frequency-hopping as example Conditioned upon in a time slot is Analysis (cont.d) Φ, the probability of successful transmission Expanding the aggregated interference leads to
Analysis (cont.d) Continuing the analysis with frequency-hopping Considering, we obtain The probability generating functional (PGFL) of PPP with intensity λ is By applying the PGFL to D(N), we obtain the mean local delay in the frequency-hopping case Analysis for ALOHA is analogous
Analytical results Frequency-hopping Mean Local Delay ALOHA Comments Infinite mean LD when N=1 or p=1, i.e., without frequencyhopping/aloha If p=1/n, the only difference between the two cases is thermal noise If frequency-hopping pattern is static, the mean LD is still infinite for any N; that is, the reduction of LD comes from reducing interference correlation, rather than reducing net amount of interference
Why does mean LD diverge when N=1? Distribution of LD is the conditional mean number of time slots for a successful transmission, and is itself a R.V. with support (1,+ ) When N=1, we bound the tail (ccdf) of this conditional mean delay as
Distribution of LD (cont.d) Lower bound for ccdf of the conditional mean delay (N = 1) θ=1 θ=10 θ=100 ccdf 10-1 Heavy tail! 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Number of divided frequency bands (N) When frequency splitting is applied (N>1), there is an additional term (N 1)/N in the success probability, preventing it from getting too small when x 0.
Variance Analysis of LD Frequency-hopping The total delay of sending a packet is, where is the delay of transmitting the ith segment The variance of delay is By applying the full expectation formula, we have The value is evaluated by applying the PGFL of PPP
Variance of Local Delay Analytical results Frequency-hopping ALOHA Comments Infinite variance when N=1 or p=1, i.e., without frequencyhopping/aloha Even with noise ignored and p=1/n set, there is still fundamental difference between the variances of two MAC schemes
Comparison 300 250 Variance of Delay 200 150 100 50 FH, theta = 1 FH, theta = 10 FH, theta = 100 ALOHA, theta = 1 ALOHA, theta = 10 ALOHA, theta = 100 Frequency-hopping: 0 5 10 15 20 25 30 35 Number of frequency bands (N) ALOHA:
Mean-Variance Tradeoff Frequency-hopping 30 25 theta = 1 theta = 10 theta = 100 Favorable operating regime Variance of LD V(N) 20 15 10 5 0 0 2 4 6 8 10 12 14 16 18 20 Mean LD D(N)
Mean-Variance Tradeoff ALOHA 300 250 theta = 1 theta = 10 theta = 100 Favorable operating regime Variance of LD V(1/N) 200 150 100 50 0 0 2 4 6 8 10 12 14 16 18 20 Mean LD D(1/N)
Mean local delay as function of N Optimal Parameters 50 50 50 45 45 45 40 40 40 35 35 35 Local Delay (D) 30 25 20 Local Delay (D) 30 25 20 Local Delay (D) 30 25 20 15 15 15 10 10 10 5 0 θ=1 θ=10 θ=100 10 20 30 40 50 60 70 80 90 100 Number of frequency bands (N) 5 0 θ=1 θ=10 θ=100 10 20 30 40 50 60 70 80 90 100 Number of frequency bands (N) 5 0 θ=1 θ=10 θ=100 10 20 30 40 50 60 70 80 90 100 Number of frequency bands (N) Optimal number of sub-bands and optimal SINR threshold exist to minimize the mean local delay
Frequency-hopping Optimal Degrees of Randomization Bounds on the optimal number of sub-bands Special Case: when ALOHA Bounds on the optimal transmitting probability
Optimal Parameters (cont.d) Bounds provide good approximation to optimal parameters Optimal number of frequency sub-bands N opt 70 60 50 40 30 20 θ=1 θ=10 θ=100 Minimum local delay D(N opt ) 40 35 30 25 20 15 10 θ=1 θ=10 θ=100 10 5 0 2 2.5 3 3.5 4 Path loss exponent α Optimal number of sub-bands and its bounds 0 2 2.5 3 3.5 4 Path loss exponent α Min value of local delay
Optimal SINR Threshold Frequency-hopping Noise-limited Interference-limited Optimal Parameters (cont.d) ALOHA Noise-limited Interference-limited
Wrap-up Without MAC randomization, the network local delay has a heavy tail distribution which results in infinite mean and variance Interference correlation is the key issue, instead of the net amount of interference Randomized channel access schemes like frequency-hopping and ALOHA rescue the local delay FH and ALOHA essentially equivalent in terms of mean LD (with p = 1/N) But FH much more favorable when considering variance of LD Parameters like number of sub-channels, transmitting probability, and SINR threshold would better be appropriately tuned