7. Marco Referencial
7.1. Marco Conceptual
7.1.5. Victimologia
(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.
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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 LATEX† by the author.
†LATEX is a document preparation system developed by Leslie Lamport as a special