A simple direct solution to a storage allocation model

Proceedings of the 6th International Conference on Queueing Theory and Network Applications

2011

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We consider models of queue storage, where items arrive accordingly to a Poisson process of rate λ and each item takes up one cell in a linear array of cells, which are numbered {1, 2, 3, · · · }. The arriving item is placed in the lowest numbered available cell. The total service rate provided to the items is the constant μ (with ρ = λ/μ), but service may be provided simultaneously to more than one item. If there are N items stored and each is serviced at the rate μ/N , this corresponds to processor sharing (PS). We analyze two models of this type, which have been shown to provide bounds on the PS model. We shall assume that (1) the server works only on the left-most item, or (2) on the two rightmost items. The set of occupied cells at any time is {i1, i2, · · · , iN } where i1 < i2 < · · · < iN and we are interested in the wasted space (iN − N ), the maximum occupied cell (iN ), and the joint distribution of the wasted space and the number of items in the system (N ). We study these exactly and asymptotically, especially in the heavy traffic limit where ρ ↑ 1.

Section: Introductionmentioning

confidence: 99%

On some simple single server models of dynamic storage

Proceedings of the 6th International Conference on Queueing Theory and Network Applications

2011

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“…A related model, the M/M/∞ queue with ranked servers, has been studied by many authors [1,3,5,[10][11][12][13][14]. This differs from the current model in that if there are a total of N = N 1 +N 2 spaces occupied, the total service rate is μN , as each customer in the restaurant is served at rate μ.…”

mentioning

confidence: 98%

“…For the M/M/∞ model a simple derivation of the exact joint distribution of finding N 1 (resp., N 2 ) occupied primary (resp., secondary) spaces appears in [13] and detailed asymptotic results for this joint distribution appear in [8,9], while the distribution of Max S is analyzed in [7].…”

mentioning

confidence: 99%

Storage allocation under processor sharing I: exact solutions and asymptotics

2010

Queueing Syst

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We consider a processor-sharing storage allocation model, which has m primary holding spaces and infinitely many secondary ones, and a single processor servicing the stored items (customers). An arriving customer takes a primary space, if one is available. We define the traffic intensity ρ to be λ/μ where λ is the customers' arrival rate and μ is the service rate of the processor. We study the joint probability distribution of the numbers of occupied primary and secondary spaces. For 0 < ρ < 1, we obtain the exact solutions for m = 1 and m = 2. For arbitrary m we study the problem in the asymptotic limit ρ ↑ 1 with m fixed. We also give the tail of the distribution for a fixed ρ < 1 and any m.

“…+ 2 = ), the pair ( 1 , 2 ) is a Markov chain whose joint steady state distribution can be explicitly calculated (see Refs [18,19]. for some recent simplified approaches to this problem).…”

mentioning

confidence: 99%

Some Exact and Asymptotic Solutions to Single Server Models of Dynamic Storage

2012

Stochastic Models

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We consider models of queue storage, where items arrive accordingly to a Poisson process of rate and each item takes up one cell in a linear array of cells, which are numbered 1, 2, 3,. The arriving item is placed in the lowest numbered available cell. The total service rate provided to the items is the constant (with = / ), but service may be provided simultaneously to more than one item. If there are items stored and each is serviced at the rate / , this corresponds to processor-sharing (PS). We analyze several models of this type, which have been shown to provide bounds on the PS model. We shall assume that (1) the server works only on the leftmost item, (2) on the two rightmost items, or (3) on all items by the PS discipline, but with a departure of the rightmost item causing a rearrangement of the items within the cells, so that the wasted space remains unchanged. The set of occupied cells at any time is i 1 , i 2 , , i where i 1 < i 2 < · · · < i and we are interested in the wasted space (W = i − ), the maximum occupied cell (i ), and the joint distribution of W and . We study these exactly and asymptotically, especially in the heavy traffic limit where ↑ 1.