Proof of Theorem 4.4.3

(Lower Bound) Note that optimization used to calculate the quantities

̂

𝐻(𝛾𝐴𭑓) and 𝐻𭑈𭐵(𝛾𝐴𭑓, 𝛾) have the same objective and same constraints except

one. Observing the fact that the every feasible point for OPT1 is also a feasible point for OPT2. We have that ,

̂

(Upper Bound) Let {𝐴*

𭑓,ℎ, 𝜇*} denote the optimizer for the convex

problem OPT2. It can be shown that the scaled point 𝛾 ∗ ({𝐴*

𭑓,ℎ, 𝜇*}) sat-

isfies all the constraints of the optimization problem to find 𝐻𭑈𭐵_(𝛾𝐴 𭑓, 𝛾),

and therefore a feasible point in the search space of optimization problem of 𝐻𭑈𭐵_(𝛾𝐴

𭑓, 𝛾). We thus have the following upper bound,

𝐻𭑈𭐵_(𝛾𝐴 𭑓, 𝛾) ≤ � 𭑓 0<ℎ<𭑁� ℎ𝛾𝐴 * 𭑓,ℎ. As {𝐴*

𭑓,ℎ, 𝜇*} was the optimizer for OPT2, we have that

̂

𝐻(𝐴_𭑓) = �

𭑓 0<ℎ<𭑁� ℎ𝐴 * 𭑓,ℎ

Using the above equality, we have that

𝐻𭑈𭐵_(𝛾𝐴

Appendix D

Proofs for Chapter 5

D.1 Proof of Theorem 5.3.1

For any given 𝜖 > 0, we will show that for all the arrival rates that belong to (1 − 𝜖)Λ, the proposed algorithm - I can stabilize the network. Let us define a quadratic Lyapunov function, 𝐿( ⃗𝑄[𝑡]) as follows,

𝐿( ⃗𝑄[𝑡]) = � 𭑙 (𝑄𭑙[𝑡]) 2_{+ �} 𭑙,𭑖,𭑗(𝑄 𭐻𭑖,𭐻𭑗 𭑙 [𝑡])2 (D.1)

Consider the drift in the Lyapunov function, Δ𝐿(.), 𝐸 [Δ𝐿( ⃗𝑄[𝑡])| ⃗𝑄[𝑡]] ∶= 𝐸 [𝐿( ⃗𝑄[𝑡 + 1]) − 𝐿( ⃗_{𝑄[𝑡])⏐⏐⏐ ⃗𝑄[𝑡]]} = 𝐸 [� 𭑙 ((𝑄𭑙[𝑡 + 1]) 2_{− (𝑄} 𭑙[𝑡])2)⏐⏐⏐_⏐⏐𝑄[𝑡]]⃗ + 𝐸 [� 𭑙,𭑖,𭑗((𝑄 𭐻𭑖,𭐻𭑗 𭑙 [𝑡 + 1])2− (𝑄𭐻𭑙 𭑖,𭐻𭑗[𝑡])2)⏐⏐⏐⏐ ⏐𝑄[𝑡]]⃗ ≤ 𝑀 + 𝐸 [� 𭑙 (2𝑄𭑙[𝑡]Δ𝑄𭑙[𝑡])⏐⏐⏐⏐⏐ ⃗ 𝑄[𝑡]] + 𝐸 [� 𭑙,𭑖,𭑗(2𝑄 𭐻𭑖,𭐻𭑗 𭑙 [𝑡]Δ𝑄𭐻𭑙 𭑖,𭐻𭑗[𝑡])⏐⏐⏐⏐ ⏐𝑄[𝑡]] ,⃗

where the last inequality follows from our assumption that the number of arrivals and departures in any time slot are bounded. Using the definition

of conditional expectation 𝐸[𝑋] = ∑_𭑦𝑃(𝑌 = 𝑦)𝐸[𝑋|𝑌 = 𝑦], we have the following, 𝐸 [Δ𝐿( ⃗𝑄[𝑡])| ⃗𝑄[𝑡]] ≤ 𝑀 + � 𭑘 𝜋(𝐻𭑘)𝐸 [�𭑙 (2𝑄𭑙[𝑡]Δ𝑄𭑙[𝑡])⏐⏐⏐⏐⏐ ⃗ 𝑄[𝑡], 𝐻[𝑡] = 𝐻𭑘] + � 𭑘 𝜋(𝐻𭑘)𝐸 [�𭑙,𭑖,𭑗(2𝑄 𭐻𭑖,𭐻𭑗 𭑙 [𝑡]Δ𝑄𭐻𭑙 𭑖,𭐻𭑗[𝑡])⏐⏐⏐⏐ ⏐𝑄[𝑡], 𝐻⃗ 𭑘] . Let us define 𝑊𭑎𭑙𭑔_{( ⃗𝑄[𝑡], 𝐻[𝑡]) ∶= max{𝑊}

𭑖𭑠[𝑡], 𝑊𭑖𭑎[𝑡]}. Using this defi-

nition, we can rewrite the above inequality as follows,

𝐸 [Δ𝐿( ⃗𝑄[𝑡])| ⃗𝑄[𝑡]] ≤ 𝑀 + 2 �

𭑙 𝑄𭑙[𝑡]𝜆𭑙− 2 �𭑘 𝜋(𝐻𭑘)𝑊

𭑎𭑙𭑔_{( ⃗𝑄[𝑡], 𝐻} 𭑘).

Since arrival rate vector lies inside Λ (i.e., ∃ 𝜖 > 0 such that ⃗𝜆 ∈ (1 − 𝜖)Λ), we can find the pair {𝑓(𝐻𭑖, 𝐻𭑗), 𝑓(𝐻𭑖), 𝛼(𝑆, 𝐻𭑖)} such that the condi-

𝐸 [Δ𝐿( ⃗𝑄[𝑡])| ⃗𝑄[𝑡]] ≤ 𝑀 − 𝜖 � 𭑙 𝑄𭑙[𝑡] + 2 � 𭑙 𝑄𭑙[𝑡] (� 𝑓(𝐻𭑖, 𝐻𭑗)𝑅𭑙(𝐻𭑖, 𝐻𭑗)) + 2 � 𭑙 𝑄𭑙[𝑡] (� 𝑓(𝐻𭑖) �𭑆 𝛼(𝑆, 𝐻𭑖)𝐼𭑙∈𭑆𝑅𭑙(𝐻𭑖)) − 2 � 𭑘 𝜋(𝐻𭑘)𝑊 𭑎𭑙𭑔_{( ⃗𝑄[𝑡], 𝐻} 𭑘).

Using the inequality that relates the 𝜋(𝐻𭑘) and 𝑓(., .), we have the

following, 𝐸 [Δ𝐿( ⃗𝑄[𝑡])| ⃗𝑄[𝑡]] ≤ 𝑀 − 𝜖 � 𭑙 𝑄𭑙[𝑡] + 2 � 𭑙 𝑄𭑙[𝑡] (� 𝑓(𝐻𭑖, 𝐻𭑗)𝑅𭑙(𝐻𭑖, 𝐻𭑗)) + 2 � 𭑙 𝑄𭑙[𝑡] (� 𝑓(𝐻𭑖) �𭑆 𝛼(𝑆, 𝐻𭑖)𝐼𭑙∈𭑆𝑅𭑙(𝐻𭑖)) − 2 � 𭑘 (�𭑗 𝑓(𝐻𭑘, 𝐻𭑗) + �𭑗 𝑓(𝐻𭑗, 𝐻𭑘) + 𝑓(𝐻𭑘)) × 𝑊𭑎𭑙𭑔_{( ⃗𝑄[𝑡], 𝐻} 𭑘).

𝐸 [Δ𝐿( ⃗𝑄[𝑡])| ⃗𝑄[𝑡]] ≤ 𝑀 − 𝜖 � 𭑙 𝑄𭑙[𝑡] + 2 � 𭑖,𭑗 𝑓(𝐻𭑖, 𝐻𭑗) (�𭑙 𝑄𭑙[𝑡]𝑅𭑙(𝐻𭑖, 𝐻𭑗) − 𝑊 𭑎𭑙𭑔_{( ⃗𝑄[𝑡], 𝐻} 𭑖) − 𝑊𭑎𭑙𭑔( ⃗𝑄[𝑡], 𝐻𭑗)) + 2 � 𭑖 𝑓(𝐻𭑖) (�𭑆 𝛼(𝑆, 𝐻𭑖) �𭑙∈𭑆𝑄𭑙[𝑡]𝑅𭑙(𝐻𭑖) − 𝑊 𭑎𭑙𭑔_{( ⃗𝑄[𝑡], 𝐻} 𭑖))

Using the fact that 𝑄𭑙[𝑡] ≤ (𝑄𭑙[𝑡] − 𝑄𭐻𭑙 𭑖,𭐻,𭑗) + + 𝑄𭐻𭑖,𭐻𭑗 𭑙 , we have the following, 𝐸 [Δ𝐿( ⃗𝑄[𝑡])| ⃗𝑄[𝑡]] ≤ 𝑀 − 𝜖 � 𭑙 𝑄𭑙[𝑡] + 2 � 𭑖,𭑗 𝑓(𝐻𭑖, 𝐻𭑗) (�𭑙 (𝑄𭑙[𝑡] − 𝑄 𭐻𭑖,𭐻𭑗 𭑙 ) + 𝑅𭑙(𝐻𭑖, 𝐻𭑗) − 𝑊𭑎𭑙𭑔( ⃗𝑄[𝑡], 𝐻𭑖)) + 2 � 𭑖,𭑗 𝑓(𝐻𭑖, 𝐻𭑗) (�𭑙 𝑄 𭐻𭑖,𭐻𭑗 𭑙 [𝑡]𝑅𭑙(𝐻𭑖, 𝐻𭑗) − 𝑊𭑎𭑙𭑔( ⃗𝑄[𝑡], 𝐻𭑗)) + 2 � 𭑖 𝑓(𝐻𭑖) (�𭑆 𝛼(𝑆, 𝐻𭑖) �𭑙∈𭑆𝑄𭑙[𝑡]𝑅𭑙(𝐻𭑖) − 𝑊 𭑎𭑙𭑔_{( ⃗𝑄[𝑡], 𝐻} 𭑖))

Since the last three quantities in the above inequality are negative for the proposed algorithm, we have that

𝐸 [Δ𝐿( ⃗𝑄[𝑡])| ⃗𝑄[𝑡]] ≤ 𝑀 − 𝜖 �

𭑙 𝑄𭑙[𝑡]

Thus, using the Foster-Lyapunov condition we have that the Markov chain ⃗𝑄 is positive recurrent.

Bibliography

[1] A. Reddy, S. Banerjee, A. Gopalan, S. Shakkottai, and L. Ying, “Wireless scheduling with heterogeneously delayed network-state information,” in

Proc. Ann. Allerton Conf. Communication, Control and Computing,

2010.

[2] L. Tassiulas and A. Ephremides, “Stability properties of constrained queue- ing systems and scheduling policies for maximum throughput in multihop radio networks,” IEEE Trans. Automat. Contr. , 1992, pp. 1936–48. [3] M. Andrews, K. Kumaran, K. Ramanan and A.L. Stolyar and R. Vi-

jayakumar and P. Whiting, “CDMA data QoS scheduling on the forward link with variable channel conditions,” Bell Labs Tech. Memo, April 2000. [4] L. Tassiulas and A. Ephremides, “Dynamic server allocation to parallel queues with randomly varying connectivity,” IEEE Trans. Inform. The-

ory, vol. 39, pp. 466–478, 1993.

[5] S. Shakkottai and A. Stolyar, “Scheduling for multiple flows sharing a time-varying channel: The exponential rule,” Ann. Math. Statist., vol. 207, pp. 185–202, 2001.

[6] A. Stolyar, “Maximizing queueing network utility subject to stability: Greedy primal-dual algorithm,” Queueing Systems, vol. 50, pp. 401–457,

August 2005.

[7] A. Eryilmaz, R. Srikant and J. Perkins, “Stable scheduling policies for fading wireless channels,” IEEE/ACM Trans. Network., vol. 13, pp. 411–424, April 2005.

[8] M. J. Neely, E. Modiano and C. E. Rohrs, “Power and server allocation in a multi-beam satellite with time varying channels,” Proc. IEEE Infocom, vol. 3, pp. 1451–1460, 2002.

[9] M. J. Neely, E. Modiano and C. Li, “Fairness and optimal stochastic control for heterogeneous networks,” Proc. IEEE Infocom, vol. 3, pp. 1723–1734, March 2005.

[10] X. Lin and S. Rasool, “Constant-Time Distributed Scheduling Policies for Ad HocWireless Networks,” Proc. Conf. on Decision and Control, 2006. [11] C. Joo and N. Shroff, “Performance of Random Access Scheduling Schemes

in Multihop Wireless Networks,” Proc. IEEE Infocom, 2007.

[12] S. Rajagopalan, J. Shin, D. Shah, “Network Adiabatic Theorem: An Efficient Randomized Protocol for Contention Resolution,” Proc. Ann.

ACM SIGMETRICS Conf., 2009.

[13] L. Jiang and J. Walrand, “A Distributed CSMA Algorithm for Through- put and Utility Maximization in Wireless Networks,” Proc. Ann. Allerton

[14] J. Liu, Y. Yi, A. Proutiere, M. Chiang and H. V. Poor, “Maximizing utility via random access without message passing,” Microsoft Research

Technical Report, 2008.

[15] S. Rajagopalan, D. Shah and J. Shin, “Network adiabatic theorem: An efficient randomized protocol for contention resolution,” Proc. Ann. ACM

SIGMETRICS Conf., pp. 133–144, 2009.

[16] J. Ni, B. Tan and R. Srikant, “Q-CSMA: Queue-length based CSMA/CA algorithms for achieving maximum throughput and low delay in wireless networks,” Proceedings of IEEE INFOCOM Mini-Conference, 2010. [17] S. Ahmad, L. Mingyan, T. Javidi, Q. Zhao and B. Krishnamachari, “Op-

timality of Myopic Sensing in Multichannel Opportunistic Access,” Infor-

mation Theory, IEEE Transactions on, vol. 55, pp. 4040–4050, 2009.

[18] S. Guha, K. Munagala and S. Sarkar, “Performance Guarantees Through Partial Information Based Control in Multichannel Wireless Networks,”

http://www.seas.upenn.edu/∼swati/report.pdf, 2006.

[19] N. Chang and M. Liu, “Optimal channel probing and transmission schedul- ing for opportunistic spectrum access,” ACM Int. Conf. on Mobile Com-

puting and Networking (MobiCom), 2007.

[20] A. Gopalan, C. Caramanis and S. Shakkottai, “On Wireless Scheduling with Partial Channel State Information,” IEEE Trans. Inform. Theory, 2011.

[21] P. Chaporkar, A. Proutiere, H. Asnani, and A. Karandikar, “Scheduling with limited information in wireless systems,” in ACM Mobihoc, 2009, pp. 75–84.

[22] A. Pantelidou, A. Ephremides, and A. Tits, “Joint scheduling and routing for ad-hoc networks under channel state uncertainty,” in Intl. Symposium

on Modeling and Optimization in Mobile, Ad-Hoc and Wireless Networks (WiOpt), April 2007, pp. 1–8.

[23] K. Kar, X. Luo and S. Sarkar, “Throughput-optimal Scheduling in Multi- channel Access Point Networks under Infrequent Channel Measurements,”

Proceedings of IEEE Infocom, 2007.

[24] J. Chen, R. A. Berry, and M. L. Honig, “Limited feedback schemes for downlink OFDMA based on sub-channel groups,” IEEE J. Sel. Areas

Commun., vol. 26, pp. 1451–1461, 2008.

[25] M. Ouyang and L. Ying, “On scheduling in multi-channel wireless down- link networks with limited feedback,” Proc. Ann. Allerton Conf. Com-

munication, Control and Computing, pp. 455- 469, 2009.

[26] M. Ouyang and L. Ying, “On Optimal Feedback Allocation in Multichan- nel Wireless Downlinks,” ACM Mobihoc, 2010.

[27] L. Ying and S. Shakkottai, “On Throughput Optimality with Delayed Network-State Information,” Technical Report, 2008.

[28] L. Ying and S. Shakkottai, “Scheduling in Mobile Ad Hoc Networks with Topology and Channel-State Uncertainty,” Proc. IEEE Infocom, pp. 2347–2355, 2009.

[29] P. Billingsley, “Probability and Measure,” Wiley Interscience Publication, 1994.

[30] S. Asmussen, “Applied Probability and Queues,” Springer-Verlag, New York, 2003.

[31] B. Birand, M. Chudnovsky, B. Ries, P. Seymour, G. Zussman, and Y. Zwols, “ Analyzing the performance of greedy maximal scheduling via local pool- ing and graph theory,” In Proc. IEEE Infocom., San Diego, California, March 2010.

[32] J. G. Dai, “ On positive harris recurrence of multiclass queueing net- works: A unified approach via fluid limit models,” The Annals of Applied

Probability, 5(1):49–77, 1995.

[33] A. Dimakis and J. Walrand, “ Sufficient conditions for stability of longest- queue-first scheduling: Second-order properties using fluid limits,” Ad-

vances in Applied Probability, 38(2):505–521, 2006.

[34] L. Georgiadis, M.J. Neely, and L. Tassiulas, “ Resource allocation and cross-layer control in wireless networks,” Foundations and Trends in Net-

[35] C. Joo, X. Lin, and N. B. Shroff, “ Understanding the capacity region of the greedy maximal scheduling algorithm in multi-hop wireless networks,”

IEEE/ACM Transactions on Networking, 17(4):1132–1145, August 2009.

[36] C. Joo and N. B. Shroff, “ Performance of random access scheduling schemes in multihop wireless networks,” In Proc. IEEE Infocom., 2007. [37] L. Bao Le, E. Modiano, C. Joo, and N. B. Shroff, “ Longest-queue-first

scheduling under sinr interference model,” In Proceedings of the eleventh

ACM international symposium on Mobile ad hoc networking and comput- ing, MobiHoc ’10, pages 41–50, 2010.

[38] B. Li, C. Boyaci, and Y. Xia, “ A refined performance characterization of longest-queue-first policy in wireless networks,” In Proceedings of the

tenth ACM international symposium on Mobile ad hoc networking and computing, MobiHoc ’09, pages 65–74, New York, NY, USA, 2009.

[39] Q. Li and R. Negi, “ Greedy maximal scheduling in wireless networks,” In GLOBECOM’10, pages 1–5, 2010.

[40] X. Lin and S. Rasool, “Constant-time distributed scheduling policies for ad hocwireless networks,” In Proc. Conf. on Decision and Control, 2006. [41] A. Reddy, S. Sanghavi, and S. Shakkottai, “ On the effect of channel fading on greedy scheduling,” In Proceedings of IEEE Infocom, Orlando, FL, 2012.

[42] L. Tassiulas and A. Ephremides, “ Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks,” IEEE Trans. Automat. Contr., 4:1936–1948, December 1992.

[43] G. Zussman and A. Brzezinski; E. Modiano;, “ Multihop local pooling for distributed throughput maximization in wireless networks,” In Proc.

IEEE Infocom., Phoenix, Arizona, April 2008.

[44] M. Jain, J. I. Choi, T. Kim, D. Bharadia, S. Seth, K. Srinivasan, P. Levis, S. Katti, and P. Sinha, “ Achieving single channel, full duplex wireless communication,” In ACM Int. Conf. on Mobile Computing and Net-

working (MobiCom), New York, USA, 2011.

[45] L. X. Bui, R. Srikant, and A. Stolyar, “ A novel architecture for re- duction of delay and queueing structure complexity in the back-pressure algorithm,” IEEE/ACM Transactions on Networking, 19:1597–1609, De- cember 2011.

[46] J. I. Choi, M. Jain, K. Srinivasan, P. Levis, and S. Katti, “ Achieving single channel, full duplex wireless communication,” In ACM Int. Conf.

on Mobile Computing and Networking (MobiCom), New York, USA, 2010.

[47] W. J. Dally and C. L. Seitz, “The torus routing chip,” Distributed Com-

[48] A. Gupta, X. Lin, and R. Srikant, “Low-complexity distributed schedul- ing algorithms for wireless networks,” In Proc. IEEE Infocom., 2007. [49] M. Jain, J. I. Choi, K. Srinivasan, S. Seth, P. Levis, and S. Katti, “ Single

channel, full-duplex wireless,” In Proceedings of the 9th ACM Conference

on Embedded Networked Sensor Systems, pages 357–358, New York, NY,

USA, 2011.

[50] L. Jiang and J. Walrand, “A distributed csma algorithm for throughput and utility maximization in wireless networks,” In Proc. Ann. Allerton

Conf. Communication, Control and Computing, 2008.

[51] C. Joo, X. Lin, and N. B. Shroff, “ Understanding the capacity region of the greedy maximal scheduling algorithm in multi-hop wireless networks,”

IEEE/ACM Transactions on Networking, 17(4):1132–1145, August 2009.

[52] M. Leconte, J. Ni, and R. Srikant, “Improved bounds on the throughput efficiency of greedy maximal scheduling in wireless networks,” In Pro-

ceedings of the tenth ACM international symposium on Mobile ad hoc networking and computing, MobiHoc ’09, 2009.

[53] L. Lin, X. Lin, and N. Shroff, “ Low-complexity and distributed energy minimization in multi-hop wireless networks,” In Proc. IEEE Infocom., 2007.

[54] N. McKeown, “ Scheduling Algorithms for Input-Queued Cell Switches,” PhD Thesis, University of California at Berkeley„ Berkeley, CA, 1995.

[55] K. Miller, A. Sanne, K. Srinivasan, and S. Vishwanath, “ Enabling real- time interference alignment: promises and challenges,” In Proceedings of

the thirteenth ACM international symposium on Mobile Ad Hoc Network- ing and Computing, MobiHoc ’12, New York, NY, USA, 2012.

[56] S. Rajagopalan, J. Shin, and D. Shah, “ Network adiabatic theorem: An efficient randomized protocol for contention resolution,” In Proc. Ann.

ACM SIGMETRICS Conf., 2009.

[57] A. Reddy, S. Sanghavi, and S. Shakkottai, “ On the effect of channel fading on greedy scheduling,” In Proceedings of IEEE Infocom, Orlando, FL, 2012.

[58] A. Sahai, G. Patel, and A. Sabharwal, “Pushing the limits of full-duplex: Design and real-time implementation,” CoRR, abs/1107.0607, 2011. [59] X. Wu and R. Srikant, “ Regulated maximal matching: a distributed

scheduling algorithm for multi-hop wireless networks with node-exclusive spectrum sharing,” In Proc. Conf. on Decision and Control, 2005. [60] X. Wu and R. Srikant, “ Bounds on the capacity region of multi-hop

wireless networks under distributed greedy scheduling,” In Proc. IEEE

Infocom., 2006.

[61] L. Ying, S. Shakkottai, A. Reddy, and S. Liu, “ On combining shortest- path and back-pressure routing over multihop wireless networks,” IEEE/ACM

[62] G. Zussman, A. Brzezinski, and E. Modiano, “ Multihop local pooling for distributed throughput maximization in wireless networks,” In Proc.

IEEE Infocom., Phoenix, Arizona, April 2008.

[63] M. Maddah-Ali, A. Motahari, and A. Khandani, “Communication over mimo x channels: interference alignment, decomposition, and perfor- mance analysis,” IEEE Trans. Inform. Theory, vol. 54, pp. 3893–3908, 2008.

[64] B. Nazer, M. Gastapar, S. A. Jafar, and S. Vishwanath, “Ergodic interfer- ence alignment,” in Proc. IEEE Int. Symp. Information Theory (ISIT), 2009.

[65] V. R. Cadambe and S. A. Jafar, “Interference alignment and degrees of freedom of the k-user interference channel,” IEEE Trans. Inform.

Theory, vol. 54, pp. 3425–3441, 2008.

[66] S. W. Jeon and M. Gastpar, “A survey on interference networks: inter- ference alignment and neutralization,” in Entropy, sep. 2012.

[67] L. Georgiadis, M. Neely, and L. Tassiulas, “Resource allocation and cross- layer control in wireless networks,” Foundations and Trends in Network-

ing, vol. 1, no. 1, pp. 1–144, 2006.

[68] C. Geng and S. A. Jafar, “On optimal ergodic interference alignment,” in

[69] T. Gou, S. A. Jafar, C. Wang, S. W. Jeon, and S. Y. Chung, “Aligned interference neutralization and degrees of freedom of the 2 x 2 x 2 inter- ference channel,” IEEE Trans. Inform. Theory, vol. 58, pp. 4381–4395, 2012.

[70] L. Bui, R. Srikant and A. Stolyar, “Novel Architectures and Algorithms for Delay Reduction in Back-pressure Scheduling and Routing,” Proc. of

Vita

Akula Aneesh Reddy received the Bachelor of Technology degree in Electrical Engineering from the Indian Institute of Technology (IIT) Madras in May 2006 and M. S. E. degree in Electrical and Computer Engineering from The University of Texas at Austin in December 2008. He was awarded Young Engineering Fellowship from Indian Institute of Science (IISc) Bangalore in July 2005. He worked as summer intern at Qualcomm Corp. R&D in 2009 and at Texas Instruments in 2010. He is currently pursuing a Ph.D. under the supervision of Dr. Sanjay Shakkottai.

Permanent address: 3453 Lake Asutin Blvd Austin, Texas 78703

This dissertation was typeset with LA_TEX† _{by the author.}

†_{LATEX is a document preparation system developed by Leslie Lamport as a special}

In document On distributed scheduling for wireless networks with time-varying channels (Page 169-186)