scholarly journals Asymptotic Analysis of a Storage Allocation Model with Finite Capacity: Joint Distribution

2016 ◽  
Vol 2016 ◽  
pp. 1-56 ◽  
Author(s):  
Eunju Sohn ◽  
Charles Knessl
Keyword(s):  
Steady State ◽  
Joint Distribution ◽  
Arrival Rate ◽  
Kolmogorov Equation ◽  
Finite Capacity ◽  
Storage Allocation ◽  
Allocation Model ◽  
Singular Perturbation Analysis ◽  
Poisson Arrival ◽  
Time Period

We consider a storage allocation model with a finite number of storage spaces. There aremprimary spaces andRsecondary spaces. All of them are numbered and ranked. Customers arrive according to a Poisson process and occupy a space for an exponentially distributed time period, and a new arrival takes the lowest ranked available space. We letN1andN2denote the numbers of occupied primary and secondary spaces and study the joint distributionProb[N1=k,N2=r]in the steady state. The joint process(N1,N2)behaves as a random walk in a lattice rectangle. We study the problem asymptotically as the Poisson arrival rateλbecomes large, and the storage capacitiesmandRare scaled to be commensurably large. We use a singular perturbation analysis to approximate the forward Kolmogorov equation(s) satisfied by the joint distribution.

On a storage allocation model with finite capacity

2016 ◽  
Vol 27 (5) ◽  
pp. 738-755 ◽  
Author(s):  
EUNJU SOHN ◽  
CHARLES KNESSL
Keyword(s):  
Random Walk ◽  
Steady State ◽  
Finite Number ◽  
Poisson Process ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Finite Capacity ◽  
Storage Allocation ◽  
Allocation Model ◽  
Numerical Studies

We consider a storage allocation model with a finite number of storage spaces. There are m primary spaces that are ranked {1,2,. . .,m} and R secondary spaces ranked {m + 1, m + 2,. . .,m + R}. Items arrive according to a Poisson process, occupy a space for a random exponentially distributed time, and an arriving item takes the lowest ranked available space. Letting N1 and N2 denote the numbers of occupied primary and secondary spaces, we study the joint distribution Prob[N1 = k, N2 = r] in the steady state. The joint process (N1, N2) behaves as a random walk in a lattice rectangle. We shall obtain explicit expressions for the distribution of (N1, N2), as well as the marginal distribution of N2. We also give some numerical studies to illustrate the qualitative behaviors of the distribution(s). The main contribution is to study the effects of a finite secondary capacity R, whereas previous studies had R = ∞.

scholarly journals Asymptotic Analysis of a Loss Model with Trunk Reservation I: Trunks Reserved for Fast Traffic

2008 ◽  
Vol 2008 ◽  
pp. 1-34 ◽  
Author(s):  
John A. Morrison ◽  
Charles Knessl
Keyword(s):  
Steady State ◽  
Asymptotic Analysis ◽  
Arrival Rate ◽  
Asymptotic Approximations ◽  
Steady State Probability ◽  
Total Load ◽  
Switched Network ◽  
Poisson Arrival ◽  
Blocking Probabilities ◽  
Single Link

We consider a model for a single link in a circuit-switched network. The link has C circuits, and the input consists of offered calls of two types, that we call primary and secondary traffic. Of the C links, R are reserved for primary traffic. We assume that both traffic types arrive as Poisson arrival streams. Assuming that C is large and R=O(1), the arrival rate of primary traffic is O(C), while that of secondary traffic is smaller, of the order O(C). The holding times of the primary calls are assumed to be exponentially distributed with unit mean. Those of the secondary calls are exponentially distributed with a large mean, that is, O(C). Thus, the primary calls have fast arrivals and fast service, compared to the secondary calls. The loads for both traffic types are comparable (O(C)), and we assume that the system is “critically loaded”; that is, the system's capacity is approximately equal to the total load. We analyze asymptotically the steady state probability that n1 (resp., n2) circuits are occupied by primary (resp., secondary) calls. In particular, we obtain two-term asymptotic approximations to the blocking probabilities for both traffic types.

scholarly journals Storage allocation under processor sharing II: Further asymptotic results

2010 ◽  
Vol 22 (1) ◽  
pp. 45-82 ◽  
Author(s):  
EUNJU SOHN ◽  
CHARLES KNESSL
Keyword(s):  
Joint Probability ◽  
Arrival Rate ◽  
Service Rate ◽  
Joint Probability Distribution ◽  
Processor Sharing ◽  
Storage Allocation ◽  
Asymptotic Results ◽  
Allocation Model ◽  
Sample Paths ◽  
Single Processor

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. All of the spaces are numbered and ordered. An arriving customer takes the lowest available space. Dynamic storage allocation and the fragmentation of computer memory are well-known applications of this model. 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. We study the problem in two asymptotic limits: (1) m → ∞ with a fixed ρ < 1, and (2) ρ ↑ 1, m → ∞ with m(1-ρ) = O(1). The asymptotics yield insight into how many secondary spaces tend to be needed, and into the sample paths leading to the occupation of the two types of spaces. We show that the asymptotics lead to accurate numerical approximations.

Approximate analysis of the containment of contaminated sites prior to remediation

2000 ◽  
Vol 42 (1-2) ◽  
pp. 319-324 ◽  
Author(s):  
H. Rubin ◽  
A. Rabideau
Keyword(s):  
Steady State ◽  
Peclet Number ◽  
Dimensionless Parameter ◽  
Contaminated Sites ◽  
Unsteady State ◽  
Péclet Number ◽  
Vertical Barrier ◽  
Time Period ◽  
Barrier Performance ◽  
Vertical Barriers

This study presents an approximate analytical model, which can be useful for the prediction and requirement of vertical barrier efficiencies. A previous study by the authors has indicated that a single dimensionless parameter determines the performance of a vertical barrier. This parameter is termed the barrier Peclet number. The evaluation of barrier performance concerns operation under steady state conditions, as well as estimates of unsteady state conditions and calculation of the time period requires arriving at steady state conditions. This study refers to high values of the barrier Peclet number. The modeling approach refers to the development of several types of boundary layers. Comparisons were made between simulation results of the present study and some analytical and numerical results. These comparisons indicate that the models developed in this study could be useful in the design and prediction of the performance of vertical barriers operating under conditions of high values of the barrier Peclet number.

On the steady-state solution of the M/G/2 queue

1979 ◽  
Vol 11 (01) ◽  
pp. 240-255 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
General Solution ◽  
Laplace Transform ◽  
Queue Length ◽  
Joint Distribution ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
The Difference ◽  
Number Of Customers

The asymptotic behaviour of the M/G/2 queue is studied. The difference-differential equations for the joint distribution of the number of customers present and of the remaining holding times for services in progress were obtained in Hokstad (1978a) (for M/G/m). In the present paper it is found that the general solution of these equations involves an arbitrary function. In order to decide which of the possible solutions is the answer to the queueing problem one has to consider the singularities of the Laplace transforms involved. When the service time has a rational Laplace transform, a method of obtaining the queue length distribution is outlined. For a couple of examples the explicit form of the generating function of the queue length is obtained.

Mixture copulas and insurance applications

2018 ◽  
Vol 12 (2) ◽  
pp. 391-411
Author(s):  
Maissa Tamraz
Keyword(s):  
Joint Distribution ◽  
Stable Distribution ◽  
Random Variable ◽  
Fixed Time ◽  
Collective Model ◽  
Data Sets ◽  
New Model ◽  
Time Period ◽  
Insurance Data ◽  
Dependence Property

AbstractIn the classical collective model over a fixed time period of two insurance portfolios, we are interested, in this contribution, in the models that relate to the joint distributionFof the largest claim amounts observed in both insurance portfolios. Specifically, we consider the tractable model where the claim counting random variableNfollows a discrete-stable distribution with parameters (α,λ). We investigate the dependence property ofFwith respect to both parametersαandλ. Furthermore, we present several applications of the new model to concrete insurance data sets and assess the fit of our new model with respect to other models already considered in some recent contributions. We can see that our model performs well with respect to most data sets.

Hitting time in an M/G/1 queue

1999 ◽  
Vol 36 (03) ◽  
pp. 934-940 ◽  
Author(s):  
Sheldon M. Ross ◽  
Sridhar Seshadri
Keyword(s):  
Service Time ◽  
Time Distribution ◽  
Arrival Rate ◽  
Hitting Time ◽  
Service Time Distribution ◽  
Expected Time ◽  
Efficient Simulation ◽  
Simulation Procedure ◽  
Poisson Arrival ◽  
Stationary Delay

We study the expected time for the work in an M/G/1 system to exceed the level x, given that it started out initially empty, and show that it can be expressed solely in terms of the Poisson arrival rate, the service time distribution and the stationary delay distribution of the M/G/1 system. We use this result to construct an efficient simulation procedure.

Measurement of pulmonary blood flow during exercise using nitrous oxide

1962 ◽  
Vol 17 (4) ◽  
pp. 579-586 ◽  
Author(s):  
Margaret R. Becklake ◽  
C. J. Varvis ◽  
L. D. Pengelly ◽  
S. Kenning ◽  
M. McGregor ◽  
...  
Keyword(s):  
Blood Flow ◽  
Steady State ◽  
Dilution Technique ◽  
Fick Principle ◽  
Pulmonary Capillary Blood Flow ◽  
Time Period ◽  
Pulmonary Capillary ◽  
Wide Range ◽  
Pulmonary Capillary Blood ◽  
The Difference

Pulmonary capillary blood flow (Qc) in the exercising subject was calculated from the rate of disappearance of N2O during steady state breathing of an N2O-He-O2 mixture. Measurements were made after alveolar rinsing (reciprocal of N2 washout) had occurred, and up to 30 sec, a time period accompanied by minimal recirculation, since FaNN2O during this period did not rise significantly. Repeatability of the method, judged as the difference of a second estimate from a first on the same subject, was comparable to that reported for the direct Fick technique in resting subjects (31 of 33 paired observations agreed within 20%). Results over a wide range agreed with almost simultaneous measurements by a dye dilution technique (24 of 26 paired observations agreed within 20%), and when related to pulse rate and to Vo2, were comparable to those of the other workers whose subjects were studied in a similar posture. Indeed, this technique (using the indirect Fick principle under “steady state” conditions) probably attains its greatest accuracy during exercise when other methods become less easily applicable. Submitted on December 18, 1961

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