Load balancing in processor sharing systems

Question

1. What are the contributions mentioned in the paper "Load balancing in processor sharing systems" ?

2. What is the way to determine the optimal multi-strategy?

3. What is the way to minimize the mean delay in a multi-server system?

4. What is the way to balance a multi-server system?

Accepted Answer

In this paper, the authors investigate optimal load balancing strategies for a multi-class multi-server processor-sharing system with a Poisson input stream, heterogeneous service rates, and a server-dependent holding cost per unit time.. Specifically, the authors study ( i ) the centralized setting in which a dispatcher routes incoming jobs based on their service time requirements so as to minimize the weighted mean sojourn time in the system ; and ( ii ) the decentralized, distributed non-cooperative setting in which each job, aware of its service time, selects a server with the objective of minimizing its weighted mean sojourn time in the system.. For the decentralized setting the authors show the existence of a potential function, which allows us to transform the non-cooperative game into a standard convex optimization problem.. Furthermore, the authors show the existence of an optimal policy that routes a job independently of its service time requirement.. The authors also show that the set of servers used in the decentralized setting is a subset of set of servers used in the centralized setting.. Finally, the authors compare the performance perceived by jobs in the two settings by studying the so-called Price of Anarchy ( PoA ), that is, the ratio between the decentralized and the optimal centralized solutions.. Interestingly, the authors show that the PoA for their system can be unbounded.

Accepted Answer

Depending on the values of the service rates and the holding costs per unit time, the optimal multi-strategy may not use all servers, but due to constraint (16) the authors are certain that atleast one server will be used.

Accepted Answer

Since the mean delay in a FCFS queue is directly proportional to the variability of the service time distribution (Pollaczek-Khinchin formula), an interval-based policy can minimize the overall mean delay in the system.

Accepted Answer

For a multi-server FCFS system with homogeneous service capacities it was conjectured in [9], and proved in [7], that the optimal load balancing scheme consists in assigning to each server all jobs whose processing times fall within non-overlapping, continuous intervals of processing times.

Accepted Answer

The authors say that a strategy p is an equilibrium for the individual optimization problem if for each i = 1, ..., K, each j = 1, ..., C and each queue k used by class i,E[ckTk(p)|i] = min j=1,...,K E[cjTj(p)|i]. (34)Without loss of generality, the authors can replace the equilibrium condition in (34) with the conditionE[dickTk(p)|i] = min j=1,...,K diE[cjTj(p)|i].

Accepted Answer

A strategy for a class i of customers is defined to be the probability vector (pi1, ..., piC), where pij is the probability that a class i customer goes to queue j.

Accepted Answer

Since the objective function defined in (7) depends only on the mean service time at each of the servers, the authors could also interpret that the arrival stream is composed of K classes, where jobs of different classes have different service time distributions.

Accepted Answer

As r1 ↓ η, the global optimal also tends to route everything towards server 1, but the key property is that since all servers are used, the global optimal can do this in such a way that E[NG1 ] grows more slowly than the decrease of c1, and as a consequence c1E[N G 1 ] → 0.

Accepted Answer

The subset of servers that are used in the optimal load balancing is SG = {1, . . . , j∗}, wherej∗ = sup ( j ≤ C : jXk=1√ cjrj >jXk=1rk − η !rcj rj) (17)Under the optimal multi-strategy, the load on server j ∈ SG isρ∗j = 1− rcj rj P k∈SG rk − ηP k∈SG √ ckrk .

Accepted Answer

(1)From queueing theory the authors know that server j is stable if ρj(p) < 1. The authors shall say that p is a stable multi-strategy if all servers are stable.

Accepted Answer

By Little’s law, minimizing the weighted mean number of jobs is equivalent to minimizing the weighted mean sojourn time in the system.

Accepted Answer

From (5) and Proposition 2, the authors can conclude that an optimal load balancing policy is obtained by applying (13) to the solution of the following Reduced Mathematical Program (RMP):minimize X j∈S fj(ρj) (14) subject to 0 ¹ ρ ≺ 1; (15)X j∈S rjρj = η. (16)Constraint (16) guarantees that all incoming jobs are served.

Load balancing in processor sharing systems

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Most frequently asked questions

1. What are the contributions mentioned in the paper "Load balancing in processor sharing systems" ?

2. What is the way to determine the optimal multi-strategy?

3. What is the way to minimize the mean delay in a multi-server system?

4. What is the way to balance a multi-server system?

5. what is the equilibrium condition for the individual optimization problem?

6. What is the definition of a strategy for a class i customer?

7. What is the simplest way to determine the arrival time at a queue?

8. What is the key property of the global optimal?

9. What is the optimal load balancing policy?

10. what is the p if all servers are stable?

11. What is the simplest way to determine the number of jobs in a queue?

12. What is the way to determine the optimal load balancing policy?

Citations

A game-theoretic approach to computation offloading in mobile cloud computing

Generalized Nash Equilibria for the Service Provisioning Problem in Multi-Cloud Systems

A game theoretic formulation of the service provisioning problem in cloud systems

Analysis of Randomized Join-the-Shortest-Queue (JSQ) Schemes in Large Heterogeneous Processor-Sharing Systems

Distributed Dynamic Speed Scaling

References

The effect of local scheduling in load balancing designs

Performance of server selection algorithms for content replication networks

IN2000-39 Braess-like Paradoxes in Distributed Performance Optimization

Selfish traffic allocation for server farms

Individual versus Social Optimization in the Allocation of Customers to Alternative Servers

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