1. ISIT 2006 — 1 On Capture in Random-Access Systems ______________________________________________ This work was supported by the Office of Naval Research. Gam D. Nguyen Information Technology Division Naval Research Laboratory Anthony Ephremides Electrical and Computer Engineering Department and Institute for Systems Research University of Maryland Jeffrey E. Wieselthier Information Technology Division Naval Research Laboratory ISIT 2006 — July 2006

2. ISIT 2006 — 2 Random-Access System Collision channel  no capture General Multiple-Access channel  all users “succeed” In-between: Reception in the presence of interference  SINR-based model  One or more users can be successful Receiver

3. ISIT 2006 — 3 Capture Probability Capture probability:  C n = Pr{at least one transmission is successful | n simultaneous transmissions} Expected number of successful packets in a slot:  S n = E{number of successful packets | n simultaneous transmissions} Multi-Packet reception capability  Depends on detector Receiver

4. ISIT 2006 — 4 SINR-based Capture Model A packet from user j is successful if and only if b = 0: Perfect capture single detector: largest always successful multiple detectors: all are successful b = ∞: No capture (collision channel) when 2 or more transmit, none are successful P(j) = Power at receiving node from user j b = Threshold that depends on many system parameters (increasing function of rate) Receiver j

5. ISIT 2006 — 5 Earlier Work (Zorzi & Rao, JSAC 1994) t = test user P n (r 0 ) = Pr{SINR(t) > b | r t = r 0 } h(r 0 ) = pdf of r 0 (distance of user to base station)  ( * ) is not valid for b < 1  Implicitly assumes only one signal can satisfy SINR Example: Propagation loss factor  = 4 Fading and shadowing are present  which exceeds 1 when (*)(*)

6. ISIT 2006 — 6 Extend Model to Accommodate All Values of b Observations More than one user can satisfy SINR > b when b < 1  Interesting case C n = Pr{one or more users satisfy SINR condition} = Pr{largest signal satisfies SINR condition} Let user M be the one with the largest signal  Thus, C n = Pr{SINR(M) > b} Since all users are equally likely to be the largest  C n = n Pr{SINR(1) > b, M = 1}

7. ISIT 2006 — 7 Analytical Evaluation of Capture Probability C n = n Pr{SINR(1) > b, M = 1} = Pr{SINR(M) > b}  where M is the user with largest received power Example:  For In general, where F P is the common cdf of the received power levels (which are i.i.d.)

8. ISIT 2006 — 8 Simulation is Needed to Evaluate C n Users uniformly distributed in disk of radius  = 1  No fading or shadowing:  any propagation model can be accommodated Results for b > 1 are same as those obtained by others The model is not realistic!  Valid only in far-field region  Received power approaches ∞ as r approaches 0

9. ISIT 2006 — 9 More-Realistic Physical Model Assume users are uniformly distributed in a circular region of radius  = 10. No fading. Curves for C n are drastically different from those for  Previously described performance is not correct  Overestimates received power when transmitter is close to receiver  = 2  = 4  = 2 instead of

10. ISIT 2006 — 10 Multi-Packet Reception All packets for which SINR > b are successful  Not only the largest S n = n Pr{SINR(1) > b}  = 10

11. ISIT 2006 — 11 A Network with Two Destinations Users uniformly distributed throughout union of 2 circles of radius  One destination receiver in each circle  Separated by distance d Traffic distribution  Each packet has a specific destination (receiver)  Does not add to throughput when decoded at “wrong” receiver  Adds to interference at both receivers  In intersection of 2 circles  Packet is equally likely to be intended for D 1 or D 2  In rest of region  Packet is intended for closer destination

12. ISIT 2006 — 12 C n for Two-Destination Network d = 20 (circles just touching) d = 15 (circles overlap)  = 10 n 1 = n 2 = n (i.e., same number of packets for each destination)  Results demonstrate impact of “broader interference” effect resulting from overlapping user populations

13. ISIT 2006 — 13 Conclusions  Clarified a small, but important, point in capture probability evaluation  Corrected misleading statements in literature  Adjusted the propagation model  Path toward analysis of multiple-destination MAC