ASSIGNMENT, DISTRIBUTION AND QOS PROVISIONING IN COMMUNICATION NETWORKS

Facility Location Theory [8]

Underlying model in the analysis of many of the combinatorial optimization network problems, also known as Location analysis

In general, given a collection of potential facility (source) sites where the facility can be opened, and a set of demand points that must be serviced, the problem is to find the subset of , to minimize a user defined metric.

Several costs and constraints can be associated with the model for e.g. the cost of opening the facilities, the cost due to delays incurred in service, the capacities of facilities, the capacities of the links etc.

The problem is NP Hard to solve optimally, therefore has to be reduced

Various variants of the problem have been investigated

Uncapacitated Facility Location Problem

Given

a distance function d :

a cost function f

Find a subset of , that minimizes

No Constraints on capacities of the facilities and the links between demand and supply are assumed

The Metric Uncapacitated Facility Location Model

In this version, the clients and the facilities are located in a metric space, and satisfy the triangle inequality

Connected Facility Location Problem [3]

Given a graph , some potential facility nodes and some client nodes

The additional constraint is that the open facilities must be connected through a Steiner Tree

The solution opens some facilities from , assigns each client to an open facilityand connects the facilities by a tree T , minimizing

Centralized Data Placement [1]

Target – Efficient distribution of internet traffic by replicating data and caching it at several locations

The problem is where to place the replicated data, in order to serve the demands with maximum performance

The problem can also be seen as a special case of the capacitated facility location problem [3]

The problem is an extension of the data placement problem mentioned in [2], described in next slide

Data Placement Problem mentioned in [2]

Given

a set of caches , a set of clients and a universe of data objects

Each cache has a capacity , each user a demand for a particular data object

Each user has to be assigned a cache, taking into account the storage and access cost

The goals is to find the placement of data objects to caches respecting the capacities of caches and minimizing the costs incurred

Formulation

A centralized server will decide the data placement scheme. The central server will manage the routers

Access routers are connected via an undirected graph. The edges of the graph represent the links. Demand nodes are placed behind the vertices

The servers have caches installed on them. Caches have specific capacities.

Users demand data, the requests are forwarded to the access routers and if the data is found in the cache, it is served. Otherwise it is fetched from another connected access router.

Three types of costs are considered

1. Transmission delays

2. Time needed to process the requests on the cache serves

3. The price charged for installation and storage of data object in a cache server

Notation Used

The Objective function

is minimized with respect to the following constraints

Solution

The problem is NP hard, and the objective function is quadratic

Two decomposition based solutions are proposed

- Lagrangian relaxation

- Randomized rounding

Directions for future works

Decentralized implementation of solution algorithms – in either a semi centralized or a fully decentralized way

Or to propose a decentralized algorithmic solution to the problem

Coded Caching [4]

To improve the performance gains of the cache networks

The coding gains are achieved at the cost of large delivery delays

Coded caching can perform better than uncoded caching

How much coded caching gain can be achieved provided a restriction on delivery delays?

The tradeoff between coded caching and delivery delay is investigated

Formulation

An origin server is connected to a network of k edge caches. The server stores a collection of video, split into a number of symbols

To each edge caches are connected a number of users , and each user can connect to only one cache

Each cache prefetches every symbol independently with probability p , such that the memory constraints are satisfied

The server knows which symbols are stored in which caches. The users attached to the caches issue a sequence of requests for one content symbol

The server responds to these request by sending multicast packets to all k edge caches associated with it

The cached symbols can be used to create coded multicast opportunities, where a single coded packet is beneficial to more than one demand

To test the ideas, a video streaming prototype is developed that uses coded caching approach

Resource Allocation in a Data Center[5]

Formulation

Consider a data center with a set of heterogeneous servers

The incoming service requests are distributed to the server with probability

‘Time of use’ pricing policy is assumed in power distribution to the servers

Time of the day is divided in slots, indexed by , with duration , and appropriate pricing is assumed for each slot duration

Request arrival follows a poisson process, with rate in the time slot

Provided the request arrival rate, the energy pricing function, specifications of the battery array and data center, a request allocation scheme for the servers and a charging discharging scheme for the batteries is proposed

Pose the problem as a convex optimization problem and solve for the required parameters

Notations used: share of resources allocated by the server in the time

period, the probability that a service request is dispatched to server in the time slot

denotes the average processing rate of the server, is the indicator whether server is turned on ( = 1) or off in

the time slot

Optimal Data Placement on Networks With Fixed Number of Clients [6]

A variant of the data placement problem

Given the set of available objects and preference for each object of each client, decide a replication scheme for the placement of data on the local caches such that the total access costs among all clients and objects are minimized

The algorithm finds optimal placement in linear time when the object lengths are uniform

Resource Placement in Distributed Network Topologies [7]

The objective is to place the resources in the regions to minimize the cost occurred in meeting the demands

The possible practical applications are peer supported video demand services, cloud based services

The challenge is to meet the arbitrary multidimensional demand

Formulation

The system consists of k areas numbered by 1,2,3…k.

In every one can place multiple resources of different types numbered 1..m

Assume that area is associated with placement . The placement L is feasible only if storage constraints are met

Consider a stochastic demand reflecting the demand at peak hours. Let represent the number of requests for type resource in area

The statistics are calculated by an external data base

If a request is made is an area , and is assigned a resource in the placement L, then it is satisfied with cost If it assigned to a remote source, then the cost incurred is If it is not served at all, the cost is

The system is operated in two stages

1. Placement Stage: Given the demand distributions {N}, {D}, the service cost parameters {C}, area storage parameters {S}, and a matching algorithm M, optimal placement of resource that minimizes the expected costs. This stage design depends on the assignment problem solution

2. Assignment (Matching) Stage: Given a placement L, a

demand realization {N} and the service cost parameters {C}, match the resources to the demands to minimize the service cost

High complexity problems are solved by reducing complexity, the placement problem is transformed to a min-cost flow problem

References

[1] Drwal, Maciej, Jozefczyk, Jerzy, “Decomposition algorithms for data placement problem based on Lagrangian relaxation and randomized rounding” in Annals of Operations Research, 2014

[2] Chaitanya Swamy, Rajmohan Rajaraman and Ivan Baev, “Approximation Problems for Data Placement Networks” [2008]

[3] Chaitnaya Swamy and Amit Kumar, “Primal-Dual Algorithms for Connected Facility Location Problems”, 2004

[4] “Coded Caching for Delay-Sensitive Content” 2014

[5] Shuang Chen; Yanzhi Wang; Pedram, M., "Resource allocation optimization in a data center with energy storage devices," Industrial Electronics Society, IECON 2014 - 40th Annual Conference of the IEEE , vol., no., pp.2604,2610, Oct. 29 2014-Nov. 1 2014

[6] Angel, Eric, Bampis, Evripidis, Pollatos, Gerasimos G., Zissimopoulos, Vassilis, “Optimal data placement on networks with constant number of clients” in Theoretical Computer Science, 2013

[7] Yuval Rochman, “Resource Placement and Assignment in Distributed network topologies”, in INFOCOM 2013

[8] Facility Location – application and theory by Drezner Ziv, Hamacher, 2002, Springer