scholarly journals A Probabilistic Approach to Growth Networks

2021 ◽  
Author(s):  
Predrag Jelenković ◽  
Jané Kondev ◽  
Lishibanya Mohapatra ◽  
Petar Momčilović
Keyword(s):  
Large Deviations ◽  
Queueing Networks ◽  
Probabilistic Approach ◽  
Logarithmic Scale ◽  
Single Server ◽  
Closed Queueing Networks ◽  
Single Class ◽  
Rate Functions ◽  
Queue Lengths ◽  
Operating Regimes

Single-class closed queueing networks, consisting of infinite-server and single-server queues with exponential service times and probabilistic routing, admit product-from solutions. Such solutions, although seemingly tractable, are difficult to characterize explicitly for practically relevant problems due to the exponential combinatorial complexity of its normalization constant (partition function). In “A Probabilistic Approach to Growth Networks,” Jelenković, Kondev, Mohapatra, and Momčilović develop a novel methodology, based on a probabilistic representation of product-form solutions and large-deviations concentration inequalities, which identifies distinct operating regimes and yields explicit expressions for the marginal distributions of queue lengths. From a methodological perspective, a fundamental feature of the proposed approach is that it provides exact results for order-one probabilities, even though the analysis involves large-deviations rate functions, which characterize only vanishing probabilities on a logarithmic scale.

Calculation of sensitivities of throughputs and realization probabilities in closed queueing networks with finite buffer capacities

1989 ◽  
Vol 21 (1) ◽  
pp. 181-206 ◽  
Author(s):  
Xi-Ren Cao
Keyword(s):  
Perturbation Analysis ◽  
Queueing Networks ◽  
Queueing Network ◽  
Theoretical Background ◽  
System Throughput ◽  
Single Server ◽  
Finite Buffer ◽  
Closed Queueing Networks ◽  
Steady State Probability ◽  
Single Class

Perturbation analysis is an efficient approach to estimating the sensitivities of the performance measures of a queueing network. A new notion, called the realization probability, provides an alternative way of calculating the sensitivity of the system throughput with respect to mean service times in closed Jackson networks with single class customers and single server nodes (Cao (1987a)). This paper extends the above results to systems with finite buffer sizes. It is proved that in an indecomposable network with finite buffer sizes a perturbation will, with probability 1, be realized or lost. For systems in which no server can directly block more than one server simultaneously, the elasticity of the expected throughput can be expressed in terms of the steady state probability and the realization probability in a simple manner. The elasticity of the throughput when each customer’s service time changes by the same amount can also be calculated. These results provide some theoretical background for perturbation analysis and clarify some important issues in this area.

Calculation of sensitivities of throughputs and realization probabilities in closed queueing networks with finite buffer capacities

1989 ◽  
Vol 21 (01) ◽  
pp. 181-206 ◽  
Author(s):  
Xi-Ren Cao
Keyword(s):  
Perturbation Analysis ◽  
Queueing Networks ◽  
Queueing Network ◽  
Theoretical Background ◽  
System Throughput ◽  
Single Server ◽  
Finite Buffer ◽  
Closed Queueing Networks ◽  
Steady State Probability ◽  
Single Class

Perturbation analysis is an efficient approach to estimating the sensitivities of the performance measures of a queueing network. A new notion, called the realization probability, provides an alternative way of calculating the sensitivity of the system throughput with respect to mean service times in closed Jackson networks with single class customers and single server nodes (Cao (1987a)). This paper extends the above results to systems with finite buffer sizes. It is proved that in an indecomposable network with finite buffer sizes a perturbation will, with probability 1, be realized or lost. For systems in which no server can directly block more than one server simultaneously, the elasticity of the expected throughput can be expressed in terms of the steady state probability and the realization probability in a simple manner. The elasticity of the throughput when each customer’s service time changes by the same amount can also be calculated. These results provide some theoretical background for perturbation analysis and clarify some important issues in this area.

Strong Approximations of Irreducible Closed Queueing Networks

1997 ◽  
Vol 29 (2) ◽  
pp. 498-522 ◽  
Author(s):  
Hanqin Zhang
Keyword(s):  
Queueing Networks ◽  
Queue Length ◽  
Rates Of Convergence ◽  
Closed Queueing Networks ◽  
Brownian Motions ◽  
Strong Approximations ◽  
Queue Lengths

A sequence of irreducible closed queueing networks is considered in this paper. We obtain that the queue length processes can be approximated by reflected Brownian motions. Using these approximations, we get rates of convergence of the distributions of queue lengths.

Closed queueing networks with blocking-a model for flexible manufacturing systems

1984 ◽  
Vol 16 (1) ◽  
pp. 9-9
Author(s):  
David D. W. Yao ◽  
J.A. Buzacott
Keyword(s):  
Flexible Manufacturing ◽  
Queueing Networks ◽  
Manufacturing Systems ◽  
Waiting Times ◽  
Queueing Systems ◽  
Priority Queue ◽  
Single Server ◽  
Closed Queueing Networks ◽  
State Behavior ◽  
Dynamic Priority

We consider a family of single-server queueing systems with two priority classes. The system operates under a dynamic priority queue discipline in which the relative priorities of customers increase with their waiting times, and which can be characterized by the urgency number. We investigate the transient as well as the steady-state behavior of the virtual waiting times of the two classes of customer as functions of the urgency number. Stochastic orderings, the joint distribution, and surprising limit results for these processes are obtained for the first time.

On First-Come First-Served Versus Random Service Discipline in Multiclass Closed Queueing Networks

1997 ◽  
Vol 11 (3) ◽  
pp. 313-326 ◽  
Author(s):  
Ronald Buitenhek ◽  
Geert-Jan van Houtum ◽  
Jan-Kees van Ommeren
Keyword(s):  
Steady State ◽  
Queueing Networks ◽  
Form Solution ◽  
Product Form ◽  
Service Discipline ◽  
Single Server ◽  
Service Time Distribution ◽  
Closed Queueing Networks ◽  
Steady State Probabilities ◽  
Closed Network

We consider multiclass closed queueing networks. For these networks, a lot of work has been devoted to characterizing and weakening the conditions under which a product-form solution is obtained for the steady-state distribution. From this work, it is known that, under certain conditions, all networks in which each of the stations has either the first-come first-served or the random service discipline lead to the same (product-form expressions for the) steady-state probabilities of the (aggregated) states that for each station and each job class denote the number of jobs in service and the number of jobs in the queue. As a consequence, all these situations also lead to the same throughputs for the different job classes. One of the conditions under which these equivalence results hold states that at each station all job classes must have the same exponential service time distribution. In this paper, it is shown that these equivalence results can be extended to the case with different exponential service times for jobs of different classes, if the network consists of only one single-server or multiserver station. This extension can be made despite of the fact that the network is not a product-form network anymore in that case. The proof is based on the reversibility of the Markov process that is obtained under the random service discipline. By means of a counterexample, it is shown that the extension cannot be made for closed network with two or more stations.

scholarly journals Analysis of Closed Queueing Networks with Batch Service

2020 ◽  
Vol 20 (4) ◽  
pp. 527-533
Author(s):  
Elena P. Stankevich ◽  
◽  
Igor E. Tananko ◽  
Vitalii I. Dolgov ◽  
◽  
...  
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Queueing Networks ◽  
Manufacturing Systems ◽  
Queueing Network ◽  
Transport Systems ◽  
Batch Service ◽  
Single Server ◽  
Closed Queueing Networks ◽  
Number Of Customers

We consider a closed queuing network with batch service and movements of customers in continuous time. Each node in the queueing network is an infinite capacity single server queueing system under a RANDOM discipline. Customers move among the nodes following a routing matrix. Customers are served in batches of a fixed size. If a number of customers in a node is less than the size, the server of the system is idle until the required number of customers arrive at the node. An arriving at a node customer is placed in the queue if the server is busy. The batсh service time is exponentially distributed. After a batсh finishes its execution at a node, each customer of the batch, regardless of other customers of the batch, immediately moves to another node in accordance with the routing probability. This article presents an analysis of the queueing network using a Markov chain with continuous time. The qenerator matrix is constructed for the underlying Markov chain. We obtain expressions for the performance measures. Some numerical examples are provided. The results can be used for the performance analysis manufacturing systems, passenger and freight transport systems, as well as information and computing systems with parallel processing and transmission of information.

On normalization constants for closed queueing networks with finite local buffers

1998 ◽  
Vol 35 (3) ◽  
pp. 600-607
Author(s):  
Ulrich A. W. Tetzlaff
Keyword(s):  
Queueing Networks ◽  
Delta Function ◽  
Partition Functions ◽  
Balance Equations ◽  
Steady State Flow ◽  
Closed Queueing Networks ◽  
Single Class ◽  
Closed Form Solutions ◽  
Flow Balance ◽  
Number Of Customers

We present new closed form solutions for partition functions used to normalize the steady-state flow balance equations of certain Markovian type queueing networks. The results focus on single class closed product form networks with state space constraints at the queueing stations. They are achieved by combining the partition function of the open network, having finite local buffers with a delta function in order to fix the number of customers in the system.

A generalization of little's law to moments of queue lengths and waiting times in closed, product-form queueing networks

1989 ◽  
Vol 26 (01) ◽  
pp. 121-133 ◽  
Author(s):  
James McKenna
Keyword(s):  
Sojourn Time ◽  
Queueing Networks ◽  
Waiting Times ◽  
Arrival Rate ◽  
Product Form ◽  
Single Server ◽  
Queue Lengths ◽  
The Mean ◽  
Expected Sojourn Time ◽  
Product Form Queueing Networks

Little's theorem states that under very general conditions L = λW, where L is the time average number in the system, W is the expected sojourn time in the system, and λ is the mean arrival rate to the system. For certain systems it is known that relations of the form E((L) l ) = λ lE((W) l ) are also true, where (L) l = L(L – 1)· ·· (L – l + 1). It is shown in this paper that closely analogous relations hold in closed, product-form queueing networks. Similar expressions relate Nji and Sji, where Nji is the total number of class j jobs at center i and Sji is the total sojourn time of a class j job at center i, when center i is a single-server, FCFS center. When center i is a c-server, FCFS center, Qji and Wji are related this way, where Qji is the number of class j jobs queued, but not in service at center i and Wji is the waiting time in queue of a class j job at center i. More remarkably, generalizations of these results to joint moments of queue lengths and sojourn times along overtake-free paths are shown to hold.

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