EE 685 presentation

Optimal Control of Wireless Networks with Finite Buffers By Long Bao Le, Eytan Modiano and Ness B. Shroff EE 685 presentation

Objective of the paper • Considers network control for wirelessnetworks with finite buffers. • Joint flow control, routing, and scheduling algorithm proposed • Aims to achieve high network utility and deterministically bounded backlogs • The tradeoff betweenbuffer size and network utility is analyzed • Scheduling/routingcomponent of the algorithmrequires ingressqueue length information (IQI) at all network nodes. • The algorithm can achieve the same utilityperformance with delayed ingress queue length information. • Numerical results reveal nearly optimal network utility with a significant reduction inqueue backlog compared to the existing algorithm in the literature.

Motivation and basic approach • Flow controllers to deterministicallybound the queue backlogs inside the network have been used. • Specifically,Lyapunov optimization and the scheduling mechanism proposed by Giaccone et al (2007)are combined to construct joint flow control, routing and scheduling algorithmsfor wireless networks with finite buffers. • The paper considersthe general setting where traffic arrival rates can be eitherinside or outside the throughput region, internal buffers in thenetwork are finite, and dynamic routing is used to achieve thelargest possible network

Contributions • A control algorithmthat achieve high networkutility and deterministically bounded backlogs for allbuffers inside the network. Moreover, these algorithmsensure that internal buffers never overflow. • Demonstration and modelling of tradeoff between buffer sizes andachievable network utility. • It has been shown that delayed ingress queue information doesnot affect the utility of the control algorithms even though there is some cost of added queue backlogs. • Via simulation based experiments, it has been demonstrated that the considered controlalgorithms perform very well in both the under and overloadedtraffic regimes. Specifically, they achieve nearlyoptimal utility performance with very low and boundedbacklogs

Problem Framework The problem is formulated for • A wireless network which is modeled as agraph G = (V;E) where V is the set of nodes and E isthe set of links. Let N and L be the number of nodes andlinks in the network, respectively. • A time-slotted system where packet arrivals and transmissions occurat the beginning of time slots of unit length. • There are multiplenetwork flows in the network each of which corresponds aparticular source-destination pair. • Arrival traffic is stored in input reservoirs and flow controllers are employed at source nodes to inject data frominput reservoirs into the network in each time slot.

System Model • Let ncbe the source node and dc be the destination node of flowc. • The buffer at the source node nc of flow c is called as an ingress buffer. All other buffers storing packets of flowc inside the network are called internal buffers. • R(c)nc (t) : the amount of traffic of flow c injected from the inputreservoir into the network at node nc in time slot t. • LetCn be the set of flows whose source node is n. • It is assumedthat ∑cϵCnR(c)n≤ Rmaxn where Rmaxn can beused to control the burstiness of admitted traffic from node ninto the network. • Rmax = max{n} {Rmaxn} • Let C denote the total number of flowsin the network.

Wireless Network Model • Each internal node maintains multiple finite buffers (one perflow) while ingress buffers at all source nodes are assumed tobe unlimited. • This assumption is justified by the fact that inmany wireless networks bufferspace is limited in contrast to buffers in ingress routers or devices that are relatively large.

Network/Queuing parameters • lc : the internal buffer size used to store packet of flow at each network node. • Q(c)n (t) : The queue length of flow cat node n at the beginning of time slot t • Q(c)dc(t) = 0 since data packets of any flow are delivered to the higher layer uponreaching the destination node. • Thecapacity of any link is one packet per time slot. • µ(c)nm(t) : the number of packets of flow c transmitted overlink (n;m) in time slot t. • µ(c)nm(t) = 1 if packet is transmitted for flow c on (n;m), • µ(c)nm(t) = 0 otherwise • µ(c)l(t) is also used instead when link (m;n) is represented as link l • Ωinn and Ωoutn : The set of incoming and outgoing links at node n. • It has been assumed that there is a loop-free route between every source-destination pairs and no traffic can be routed back • Node n will not transmit data offlow c along any link (n;m) whenever Q(c)n < 1

Queue evolution dynamics • The queue evolutions for any network node n can be written as : where R(c)n(t) = 0, for all t and n ≠ nc. • Let r(c)n(t) be the time average rate of admitted traffic for flow c at thecorresponding source node nc up to time t :

Queue evolution dynamics • The long-term time-average admitted rate for flow c is defined as • A queue for a particular flow cat node n is called strongly stable if

Throughput region with finite buffers • The maximum throughput region Λ of a wireless network with unlimited ingress and internal buffers is the set of all traffic arrival rate vectors such that there exists a network control algorithm to stabilize all individual queues in the network. • Note that when internal buffers are finite, stability corresponds to maintaining bounded backlogs in ingress buffers (since internal buffers are finite, the issue of stability does not arise there). • In this context, it is necessary to devise a control algorithm that can achieve high throughput (utility) with finite buffers.

Throughput region with finite buffers • First, let us define the �ϵ-stripped throughput region as follows: • where �(rnc)=(rn11 ,rn22 ,...,rnCC ). • (rnc∗(c)(ϵ)) is the optimal solution of the following optimizationproblem • (λnc)=(λn11 , λn22 ,..., λnCC )Tis the average traffic arrival rate vector, (.)T denotes vector transposition,g(c)(.) are increasing and concave utility functions. • We will quantify the performance of the considered control algorithms in terms of (rnc∗(c)(ϵ)) which tends to (rnc∗(c)) as ϵ → 0

Network optimization In heavy traffic regime • This is the case where all sources are constantly backlogged • Proposed method seek a balance between optimizing the total network utility and bounding total queue backlog • Following optimization problem is pursued specifically : • where rnc(c)is the time average admitted rate for flow c at node nc, ( rnc(c))=( rn11 ,rn22 ,...,rnCC ) is the time average admitted rate vector, and lc is the buffer size. • The last inequality constraint ensures that the backlogs in internal buffers are finite and bounded by lc at all times.

Network Optimization Algorithm 1 : Constantly Backlogged Sources FLOW CONTROL • Each node n injects an amount of trafficinto the network which is equal to R(c)nc(t) = x(c)nc where x(c)nc is the solution of the following optimizationproblem

Network Optimization Algorithm 1 : Constantly Backlogged Sources • ROUTING/SCHEDULING • Each link (n;m) calculates thedifferential backlog for flow c as follows: • Then, link (n;m) calculates the maximum differential backlog as follows:

Network Optimization Algorithm 1 : Constantly Backlogged Sources • ROUTING/SCHEDULING • Let S be the schedule where its l-th component Sl = 1if link l is scheduled and Sl = 0 otherwise. • The schedule S is chosen in each time slot t as follows: • Where Φis the set of all feasible schedules as determinedby the underlying wireless interference model. Any link l with Wl ≤ 0 will not be scheduled¤ • For each scheduled link, one packet of the flow that achieves themaximum differential backlog is transmitted ifthe corresponding buffer has at least one packet waiting for transmission. .

Algorithm 1 working principlesConstantly Backlogged Sources • The flow controller of this algorithm admits less traffic offlow c if the corresponding ingress buffer is more congested(i.e., large Q(c)nc ). • A larger value V results in higherachievable utility albeit at the cost of larger queue backlogs. • The routing component of this algorithm is anadaptation of the differential backlog routing of Tassiulas andEphremides to the case of finite buffers . • The schedulingrule is the well-known max-weight scheduling algorithm.

Algorithm 1 working principlesConstantly Backlogged Sources • The differential backlog ensures that internal buffersnever overflow. • The differential backlog of any sourcelink (nc;m), given by (Q(c)nc / lc) (lc-Q(c)m)bounds the backlogs of all internal buffers by lc. • The differential backlogsof all other links (n;m), given by (Q(c)nc / lc) (Q(c)n -Q(c)m)is essentially the multiplication of the standard differential backlog with the normalized ingress queue backlog Q(c)nc / lc. • Incorporating ingress queue backlog into the differential backlog prioritizes the flow whoseingress queue is more congested. This helps stabilize ingressqueues because the scheduling component of algorithm 1 stillhas the “max-weight” structure.

Theorem 1 : • Given ϵ > 0, if the internal buffer size satisfies . • Then, we have the following bounds for utility and ingress queue backlog . • where D1=C(R2max+2)+C(N-1)Rmaxlc is a finite number, • C is the number of flows in the network, • Rmax is the maximumamount of traffic injected into the network from any node • Vis a design parameter which controls the utility and backlogbound tradeoff • λmax is the largest number such that λmaxϵΛ with λ = (λ, λ, ..., λ)T is a columnvector with all element s equal to λ .

Theorem 1 : Definition of other parameters for algorithm 1 are provided as : .

Theorem 1 :Proof

Theorem 1 :Proof • Taking the expectations over the distribution of Q and summing overt ϵ{1,2,...,M}, we have .

Theorem 1 :Proof

Theorem 1 :Proof • To prove the backlog bound, we arrange the inequality (46) appropriatelyand divide both sides by M, we have .

Network Optimization Algorithm 2 : Sources with Arbitrary Arrival Rates FLOW CONTROL • Each node n injects an amount of trafficinto the network which is equal to R(c)nc(t) as the solution of the following optimizationproblem • Each node n injects an amount of trafficinto the network which is equal to R(c)nc(t) as the solution of the following optimizationproblem • Where L(c)nc(t) is the backlog of the input reservoir,A(c)nc(t) is the number of arriving packets in time slott, and Y (c)nc(t) represents “backlog” in the virtual queue which has the following queue-like evolution

Network Optimization Algorithm 2 : Sources with Arbitrary Arrival Rates • FLOW CONTROL • Virtual queue evolution has the following form • Where z(c)nc (t) are auxiliary variables which are calculatedfrom the following optimization problem • The virtual queue variables Y(c)nc are updated according to above queue evolution mechanism in every time slot.

Network Optimization Algorithm 2 : Sources with Arbitrary Arrival Rates • ROUTING/SCHEDULING • Performed according to algorithm 1 principles .

Algorithm 2 working principles • The flow controller operation can be interpreted intuitively as follows. • The auxiliary variables z(c)nc(t) plays the role of R(c)nc(t) inalgorithm 1 for the heavy traffic regime. • In fact, the optimizationproblem from which we calculate z(c)nc(t) issimilar to the one in (12) where Q(c)nc is replaced by Y (c)nc. • Hence,z(c)nc(t) represents the potential rate that would havebeen admitted if sources were backlogged. , • The virtual queue Y (c)nc(t) captures the differencebetween the potential admittedrate z(c)nc(t) and the actual admitted rateR(c)nc(t) based on whichthe flow controller determines the amount of admitted trafficR(c)nc(t) from (27). • Here, the “virtual differential backlogs [ Y(c)nc-Q(c)nc ] determines from which flow to inject data to the network

Theorem 2 :

EE 685 presentation