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Running in Circles: Packet Routing on Ring Networks by William F

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Running in Circles: Packet Routing on Ring Networks by William F Running in Circles: Packet Routing on Ring Networks by William F. Bradley Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2002 c 2002 William F. Bradley. All rights reserved Author............................................. ............................... Department of Mathematics April 26, 2002 Certified by.......................................... .............................. F. T. Leighton Professor of Applied Mathematics Thesis Supervisor arXiv:1402.2963v1 [math.PR] 12 Feb 2014 Accepted by......................................... .............................. R. B. Melrose Chairman, Committee on Pure Mathematics 2 Running in Circles: Packet Routing on Ring Networks by William F. Bradley Submitted to the Department of Mathematics on April 26, 2002, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract I analyze packet routing on unidirectional ring networks, with an eye towards establishing bounds on the expected length of the queues. Suppose we route packets by a greedy “hot potato” protocol. If packets are inserted by a Bernoulli process and have uniform destinations around the ring, and if the nominal load is kept fixed, then I can construct an upper bound on the expected queue length per node that is independent of the size of the ring. If the packets only travel one or two steps, I can calculate the exact expected queue length for rings of any size. I also show some stability results under more general circumstances. If the packets are inserted by any ergodic hidden Markov process with nominal loads less than one, and routed by any greedy protocol, I prove that the ring is ergodic. Thesis Supervisor: F. T. Leighton Title: Professor of Applied Mathematics 3 4 For my father 5 6 Contents 1 Statement of the Problem 11 1.1 TheProblem ................................... 11 1.2 What’sinthisThesis.............................. 13 1.2.1 AGeneralOverview ........................... 13 1.2.2 TheNewResults............................. 14 1.3 RingDetails.................................... 16 1.4 TheBidirectionalRing. 16 1.5 StandardizedNotation . 17 1.6 ALittleHistory.................................. 17 2 Exact Solutions 21 2.1 Introduction.................................... 21 2.2 TheOneNodeRing ............................... 21 2.3 Fixing L intheNonstandardBernoulliRing. 23 2.4 The Stationary Distribution and Consequences . ........ 24 2.4.1 The3NodeStandardBernoulliRing. 26 2.4.2 The5NodeBidirectionalRing . 26 2.5 TheEPF,SIS,CTO,andFTGProtocols . 26 2.6 TheGHPProtocol ................................ 30 2.7 N=1,L=2..................................... 32 2.8 Proof for All N .................................. 33 2.8.1 Probability of Processor i ........................ 35 2.8.2 Probability of the Other Processors . 39 2.8.3 TheBalanceEquations ......................... 43 2.9 FutureWork,andaWarning . 47 3 Bounds on Queue Length 49 3.1 Introduction.................................... 49 3.2 LowerBounds................................... 49 3.3 Load of 1/2+ ǫ .................................. 51 3.4 LyapunovLemmas ................................ 54 3.5 ErgodicityandExpectedQueueLength . 68 3.6 OtherBernoulliRings ............................. 70 3.6.1 TheBidirectionalRing. 70 3.6.2 N constant, L →∞ ........................... 70 3.6.3 L constant, N →∞ ........................... 71 3.6.4 BoundedQueueLengths. 71 7 3.7 FutureWork ................................... 71 4 Fluid Limits 73 4.1 Introduction.................................... 73 4.2 ADriftCriterionforStability . ..... 74 4.3 OurModelofPacketRouting . 77 4.4 BuildingaBoundedNorm. .. .. .. .. .. .. .. .. 80 4.5 FluidLimits.................................... 82 4.6 FutureWork ................................... 85 5 Analyticity 87 5.1 Analyticity and Absolute Monotonicity . ....... 87 5.2 LightTrafficLimits................................ 89 5.2.1 ProductFormResults .......................... 89 5.2.2 ExplicitCalculations. 91 5.3 A Class of Absolutely Monotonic Networks . ..... 94 5.4 FutureWork ................................... 99 6 Ringlike Networks 101 6.1 Introduction.................................... 101 6.2 ConvexRouting.................................. 102 6.3 Superconcentration on a Pair of Butterflies . ....... 105 6.4 DefinitionsandNotation. 106 6.5 TheSub-ButterflyConnectivityGraph . 108 6.6 Subsets of Size 2m ................................ 109 6.6.1 SomeCorollaries ............................. 111 6.7 TheGeneralCase................................. 111 6.8 TheMatchingLemma .............................. 113 6.9 Conclusion .................................... 115 A Analysis and Probability 117 A.1 MarkovChains .................................. 117 A.2 TailBounds.................................... 118 A.3 TheComparisonTheoremandDrift . 120 A.4 Uniform Integrability Facts . 122 A.5 QueueingTheory................................. 123 B A Primer on Fluid Limits 127 B.1 Introduction.................................... 127 B.2 AnAnalyticFact................................. 128 B.3 FluidLimits.................................... 129 B.4 SpecificProtocols................................ 134 B.4.1 FIFO ................................... 134 B.4.2 PriorityDisciplines. 134 8 C Fluid Limit Examples 137 C.1 TheRing ..................................... 138 C.2 LayeredNetworks................................. 139 C.3 Meta-layeredNetworks. 139 C.4 ConvexRouting.................................. 140 C.5 Generalized Kelly Networks with FIFO . 140 C.6 PrioritizingAllPaths . .. .. .. .. .. .. .. 140 C.7 FarthestToGo .................................. 141 C.8 ClosesttoOrigin ................................. 142 C.9 LongestInSystem ................................ 142 C.10ShortestInSystem ............................... 143 C.11 Nearest To Go for Re-entrant Lines . 143 C.12RoundRobin ................................... 143 C.13LeakyBuckets .................................. 144 C.14 The Utility of Adversarial Results . 144 D Fluid Limit Counterexamples 145 D.1 Nonmonotonicstability . 145 D.2 Virtual Stations and Instability . 145 D.3 NTGcanbeUnstable .............................. 145 D.4 Uniformly Generalized Kelly Networks can be Unstable . ......... 145 D.5 A Generalized Kelly Network without Immediate Feedback can be Unstable 146 D.6 FIFOcanbeUnstable .............................. 146 D.7 Adversarially Unstable, Stochastically Stable . ........... 146 D.8 Stochastically Unstable, Adversarially Stable . ........... 146 E Analytic Computing 147 E.1 Exact Information about Stationary Distributions . .......... 147 E.2 ThePayoff .................................... 149 E.3 RealWorldDetails ................................ 149 9 10 Chapter 1 Statement of the Problem 1.1 The Problem What is packet routing? In a packet routing network, we populate the nodes of a directed graph with a collection of discrete objects called packets. As time passes, these packets oc- casionally travel across edges, or depart the network. Sometimes, new packets are inserted. A typical question to ask is: what is the expected number of packets in the system? This thesis is inspired by the following packet routing problem on the ring: Suppose we have a directed graph in the form of a cycle with the edges directed clockwise. Let’s label the nodes 1 through N, where for i < N, we have a directed edge from node i to i + 1, and an additional edge from N to 1. See Figure 1-1. 1 9 2 8 3 7 4 6 5 Figure 1-1: An N = 9 node unidirectional ring We are going to analyze the network’s behavior as it evolves in time, where time is measured in discrete steps. First, we have to specify how packets enter the ring. Let us suppose that with probability p, the probability that a new packet arrives at a node on one time step. With probability 1 − p, no packet arrives. This event occurs independently at every node, on every time step. Next, we must specify how packets travel along the ring. A packet arriving at node i chooses its destination uniformly from the other N − 1 nodes. We will allow at most one packet to depart from a node in one time step. When a packet arrives at its destination, it is immediately removed; that is, a packet waiting in queue can be inserted into the ring on the same time step. Finally, if there is more than one packet at a node, we must specify 11 which packet advances next. We’ll use the Greedy Hot Potato protocol. Definition 1 In the Greedy Hot Potato protocol, packets travelling in the network have priority over packets waiting in queue. Nodes with non-empty queues always route packets. This protocol for determining packet priority is called Greedy Hot Potato because a packet being passed along the ring is a “hot potato” that never stops moving until it reaches its destination. It is “greedy” in the sense that whenever a node has the opportunity to route a packet, it always takes it. This protocol resolves all contentions over which packet gets to depart from a node. By specifying the number of packets waiting at each of the N nodes, and the destination of each packet travelling in the ring, we completely specify the state of the system, and we have a discrete-time Markov chain. Consider the number of packets in the system that need to use a node n. At most 1 packet can depart from n on each time step. Therefore, if too many new packets arrive, the system is unstable (the Markov chain is not ergodic). In practical terms,
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