Article

Bounds For Performance Characteristics; A Systematic Approach Via Cost Structures

02/1999; DOI: 10.1080/15326349808807467
Source: CiteSeer

ABSTRACT In this paper we present a systematic approach for the construction of bounds for the average cost in Markov chains with infinitely many states. The technique to prove the bounds is based on dynamic programming. Most performance characteristics of Markovian systems can be represented by the average cost for some appropriately chosen cost structure. Therefore, the approach can be used to generate bounds for relevant performance characteristics. The approach is demonstrated for the shortest queue model. It is shown how for this model several bounds for the mean waiting time can be constructed. We include numerical results to demonstrate the quality of these bounds. 1 INTRODUCTION In this paper we consider an irreducible N-dimensional Markov chain with states m = (m 1 ; Delta Delta Delta ; mN ), where each m i is an integer, and transition probabilities p(m;n). Let ß denote its equilibrium distribution (which we assume to exist). If c(m) is the cost per period in state m, then the ...

0 Bookmarks
 · 
63 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Synchronization is often necessary in parallel computing, but it can create delays whenever the receiving processor is idle, waiting for the information to arrive. This is especially true for barrier, or global, synchronization, in which every processor must synchronize with every other processor. Nonetheless, barriers are the only form of synchronization explicitly supplied in OpenMP, and they occur whenever collective communication operations are used in MPI. Many applications do not actually require global synchronization; local synchronization, in which a processor synchronizes only with those processors from or to which information or resources are needed, is often adequate. However, when tasks take varying amounts of time the behavior of a system under local synchronization is more di!cult to analyze since processors do not start tasks at the same time. We show that when the synchronization dependencies form a directed cycle and the task times are geometrically distributed with p =0 .5, then as the number of processors tends to infinity the processors are working 2 ! " 2 # 0.59% of the time. Under global synchronization, however, the time to complete each task is unbounded, increasing logarithmically with the num- ber of processors. Similar results apply for p $ =0 .5. We also present some of the combinatorial properties of the synchronization problem with geometrically distributed tasks on an undirected cycle. Nondeterministic synchronization is also examined, where processors decide randomly at the beginning of each task which neighbors(s) to synchronize with. We show that the expected number of task dependencies for random synchronization on an undirected cycle is the same as for deterministic synchronization on a directed cycle. Simulations are included to extend the analytic results. They show that more heavy-tailed distributions can actually create fewer delays than less heavy-tailed ones if the number of pro- cessors is small for some random-neighbor synchronization models. The results also show the rate of convergence to the steady state for various task distributions and synchronization graphs.
    SIAM J. Comput. 01/2010; 39:3860-3884.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider a system of K parallel queues providing different grades of service through each of the queues and serving a multiclass customer population. Service differentiation is achieved by specifying different join prices to the queues. Customers of class j define a cost function [psi]ij(ci,xi) for taking service from queue i when the join price for queue i is ci and congestion in queue i is xi and join the queue that minimizes [psi]ij(·,·). Such a queuing system will be called the “join minimum cost queue” (JMCQ) and is a generalization of the join shortest queue (JSQ) system. Non-work-conserving (called Paris Metro pricing system) and work-conserving (called the Tirupati system) versions of the JMCQ are analyzed when the cost to an arrival of joining a queue is a convex combination of the join price for that queue and the expected waiting time in that queue at the arrival epoch. Our main results are for a two-queue system.
    Probability in the Engineering and Informational Sciences 09/2004; 18(04):445 - 472. · 0.45 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper we study a production system consisting of a group of parallel machines producing multiple job types. Each machine has its own queue and it can process a restricted set of job types only. On arrival a job joins the shortest queue among all queues capable of serving that job. Under the assumption of Poisson arrivals and identical exponential processing times we derive upper and lower bounds for the mean waiting time. These bounds are obtained from so-called flexible bound models, and they provide a powerful tool to efficiently determine the mean waiting time. The bounds are used to study how the mean waiting time depends on the amount of overlap (i.e. common job types) between the machines.
    Operations Research-Spektrum 07/2001; 23(3):411-427. · 1.41 Impact Factor

Full-text (2 Sources)

View
18 Downloads
Available from
May 22, 2014