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 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.

scholarly journals Two Coupled Queues with Vastly Different Arrival Rates: Critical Loading Case

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Charles Knessl ◽  
John A. Morrison
Keyword(s):  
Customer Service ◽  
Arrival Rate ◽  
Service Discipline ◽  
Processor Sharing ◽  
Steady State Probability ◽  
Poisson Arrival ◽  
Loading Case ◽  
Queue Lengths ◽  
Number Of Customers ◽  
Comparable Magnitude

We consider two coupled queues with a generalized processor sharing service discipline. The second queue has a much smaller Poisson arrival rate than the first queue, while the customer service times are of comparable magnitude. The processor sharing server devotes most of its resources to the first queue, except when it is empty. The fraction of resources devoted to the second queue is small, of the same order as the ratio of the arrival rates. We assume that the primary queue is heavily loaded and that the secondary queue is critically loaded. If we let the small arrival rate to the secondary queue beO(ε), where0≤ε≪1, then in this asymptotic limit the number of customers in the first queue will be large, of orderO(ε-1), while that in the second queue will be somewhat smaller, of orderO(ε-1/2). We obtain a two-dimensional diffusion approximation for this model and explicitly solve for the joint steady state probability distribution of the numbers of customers in the two queues. This work complements that in (Morrison, 2010), which the second queue was assumed to be heavily or lightly loaded, leading to mean queue lengths that wereO(ε-1)orO(1), respectively.

scholarly journals Multipreconditioned GMRES for simulating stochastic automata networks

2018 ◽  
Vol 16 (1) ◽  
pp. 986-998
Author(s):  
Chun Wen ◽  
Ting-Zhu Huang ◽  
Xian-Ming Gu ◽  
Zhao-Li Shen ◽  
Hong-Fan Zhang ◽  
...  
Keyword(s):  
Steady State ◽  
Probability Distribution ◽  
Linear Systems ◽  
Iterative Methods ◽  
Communication Systems ◽  
Queueing Systems ◽  
Steady State Probability ◽  
Stochastic Automata ◽  
Automata Networks ◽  
Generator Matrices

AbstractStochastic Automata Networks (SANs) have a large amount of applications in modelling queueing systems and communication systems. To find the steady state probability distribution of the SANs, it often needs to solve linear systems which involve their generator matrices. However, some classical iterative methods such as the Jacobi and the Gauss-Seidel are inefficient due to the huge size of the generator matrices. In this paper, the multipreconditioned GMRES (MPGMRES) is considered by using two or more preconditioners simultaneously. Meanwhile, a selective version of the MPGMRES is presented to overcome the rapid increase of the storage requirements and make it practical. Numerical results on two models of SANs are reported to illustrate the effectiveness of these proposed methods.

Asymptotic Estimates for Blocking Probabilities in a Large Multi-Rate Loss Network

1997 ◽  
Vol 29 (3) ◽  
pp. 806-829 ◽  
Author(s):  
A. Simonian ◽  
J. W. Roberts ◽  
F. Théberge ◽  
R. Mazumdar
Keyword(s):  
Critical Load ◽  
Blocking Probability ◽  
Asymptotic Estimates ◽  
Switched Network ◽  
Blocking Probabilities ◽  
Low Load ◽  
Rate Loss ◽  
Load Conditions

In this paper, asymptotic estimates for the blocking probability of a call pertaining to a given route in a large multi-rate circuit-switched network are given. Concentrating on low load and critical load conditions, these estimates are essentially derived by using probability change techniques applied to the distribution of the number of occupied links. Such estimates for blocking probabilities are also given a uniform expression applicable to both load regimes. This uniform expression is numerically validated via simple examples.

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.

scholarly journals A M/M/1 Queueing Network with Classical Retrial Policy

2021 ◽  
Vol 12 (1S) ◽  
pp. 329-337
Author(s):  
S. Shanmugasundaram, Et. al.
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Queueing Network ◽  
State Probability ◽  
Queueing Model ◽  
Steady State Probability ◽  
Numerical Examples ◽  
System Length ◽  
Number Of Customers ◽  
Little's Formula

In this paper we study the M/M/1 queueing model with retrial on network. We derive the steady state probability of customers in the network, the average number of customers in the all the three nodes in the system, the queue length, system length using little’s formula. The particular case is derived (no retrial). The numerical examples are given to test the correctness of the model.

scholarly journals Degrading systems availability analysis: analytical semi-Markov approach

2021 ◽  
Vol 23 (1) ◽  
pp. 195-208
Author(s):  
Varun Kumar ◽  
Girish Kumar ◽  
Rajesh Kumar Singh ◽  
Umang Soni
Keyword(s):  
Steady State ◽  
Degradation Mechanism ◽  
Continuous Process ◽  
Mechanical Systems ◽  
Embedded Markov Chain ◽  
Steady State Probability ◽  
Opportunistic Maintenance ◽  
System Availability ◽  
Availability Analysis ◽  
Complex Mechanical Systems

This paper deals with modeling and analysis of complex mechanical systems that deteriorate with age. As systems age, the questions on their availability and reliability start to surface. The system is believed to suffer from internal stochastic degradation mechanism that is described as a gradual and continuous process of performance deterioration. Therefore, it becomes difficult for maintenance engineer to model such system. Semi-Markov approach is proposed to analyze the degradation of complex mechanical systems. It involves constructing states corresponding to the system functionality status and constructing kernel matrix between the states. The construction of the transition matrix takes the failure rate and repair rate into account. Once the steady-state probability of the embedded Markov chain is computed, one can compute the steady-state solution and finally, the system availability. System models based on perfect repair without opportunistic and with opportunistic maintenance have been developed and the benefits of opportunistic maintenance are quantified in terms of increased system availability. The proposed methodology is demonstrated for a two-stage reciprocating air compressor with intercooler in between, system in series configuration.

Analysis of Some Stochastic Inventory System in Supply Chain

2016 ◽  
pp. 124-147
Author(s):  
Bakthavachalam Rengarajan
Keyword(s):  
Steady State ◽  
Lead Time ◽  
Distribution Center ◽  
Setup Cost ◽  
Steady State Probability ◽  
Stochastic Inventory ◽  
Replenishment Policy ◽  
System States ◽  
Policy Supply ◽  
The Cost

In this chapter we consider a three echelon inventory control system which is modeled as a warehouse, single distribute and one retailer system handling a single product. A finished product is supplied from warehouse to distribution center which adopts one-for-one replenishment policy. The replenishment of items in terms of packets from warehouse to distribution center with exponential lead time having parameter µ1. Then the product is supplied from distribution center to retailer who adopts (s, S) policy. Supply to the retailer in packets of Q (= S - s) items is administrated with exponential lead time having parameter µ0. The demand at retailer node follows a Poisson with mean lambda. The steady state probability distribution of system states and the measures of system performance in the steady state are obtained explicitly. The Cost function is computed by using numerical searching algorithms, the optimal reorder points are obtained for various input parameters. Sensitivity analysis are discussed for various cost parameter such as holding cost, setup cost etc.

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