scholarly journals A Priority Queue with Many Customer Types, Correlated Arrivals and Changing Priorities

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1292
Author(s):  
Seokjun Lee ◽  
Sergei Dudin ◽  
Olga Dudina ◽  
Chesoong Kim ◽  
Valentina Klimenok
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Performance Indicators ◽  
Queueing System ◽  
Priority Queue ◽  
First Aid ◽  
Single Server ◽  
Finite Buffer ◽  
Impatient Customers ◽  
Arrival Times

A single-server queueing system with a finite buffer, several types of impatient customers, and non-preemptive priorities is analyzed. The initial priority of a customer can increase during its waiting time in the queue. The behavior of the system is described by a multi-dimensional Markov chain. The generator of this chain, having essential dependencies between the components, is derived and formulas for computation of the most important performance indicators of the system are presented. The dependence of some of these indicators on the capacity of the buffer space is illustrated. The profound effect of the phenomenon of correlation of successive inter-arrival times and variance of the service time is numerically demonstrated. Results can be used for the optimization of dispatching various types of customers in information transmission systems, emergency departments and first aid stations, perishable foods supply chains, etc.

Queues with semi-Markovian arrivals

1967 ◽  
Vol 4 (02) ◽  
pp. 365-379 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
Finite Number ◽  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
Interarrival Time ◽  
Single Server ◽  
Queue Size

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.

scholarly journals Optimization of the service strategy in a queueing system with energy harvesting and customers’ impatience

2016 ◽  
Vol 26 (2) ◽  
pp. 367-378 ◽  
Author(s):  
Alexander Dudin ◽  
Moon Ho Lee ◽  
Sergey Dudin
Keyword(s):  
Markov Chain ◽  
Queueing System ◽  
Threshold Value ◽  
Arrival Process ◽  
Sleep Mode ◽  
Single Server ◽  
Finite Buffer ◽  
Phase Type Distribution ◽  
New Strategy ◽  
Number Of Customers

Abstract A single-server queueing system with an infinite buffer is considered. The service of a customer is possible only in the presence of at least one unit of energy, and during the service the number of available units decreases by one. New units of energy arrive in the system at random instants of time if the finite buffer for maintenance of energy is not full. Customers are impatient and leave the system without service after a random amount of waiting time. Such a queueing system describes, e.g., the operation of a sensor node which harvests energy necessary for information transmission from the environment. Aiming to minimize the loss of customers due to their impatience (and maximize the throughput of the system), a new strategy of control by providing service is proposed. This strategy suggests that service temporarily stops if the number of customers or units of energy in the system becomes zero. The server is switched off (is in sleep mode) for some time. This time finishes (the server wakes up) if both the number of customers in the buffer and the number of energy units reach some fixed threshold values or when the number of energy units reaches some threshold value and there are customers in the buffer. Arrival flows of customers and energy units are assumed to be described by an independent Markovian arrival process. The service time has a phase-type distribution. The system behavior is described by a multi-dimensional Markov chain. The generator of this Markov chain is derived. The ergodicity condition is presented. Expressions for key performance measures are given. Numerical results illustrating the dependence of a customer’s loss probability on the thresholds defining the discipline of waking up the server are provided. The importance of the account of correlation in arrival processes is numerically illustrated.

scholarly journals A Queueing System with Heterogeneous Impatient Customers and Consumable Additional Items

2017 ◽  
Vol 27 (2) ◽  
pp. 367-384 ◽  
Author(s):  
Janghyun Baek ◽  
Olga Dudina ◽  
Chesoong Kim
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Optimization Problems ◽  
Markovian Arrival Process ◽  
Arrival Process ◽  
Single Server ◽  
Finite Buffer ◽  
Threshold Strategy

Abstract A single-server queueing system with a marked Markovian arrival process of heterogeneous customers is considered. Type-1 customers have limited preemptive priority over type-2 customers. There is an infinite buffer for type-2 customers and no buffer for type-1 customers. There is also a finite buffer (stock) for consumable additional items (semi-products, half-stocks, etc.) which arrive according to the Markovian arrival process. Service of a customer requires a fixed number of consumable additional items depending on the type of the customer. The service time has a phase-type distribution depending on the type of the customer. Customers in the buffer are impatient and may leave the system without service after an exponentially distributed amount of waiting time. Aiming to minimize the loss probability of type-1 customers and maximize throughput of the system, a threshold strategy of admission to service of type-2 customers is offered. Service of type-2 customer can start only if the server is idle and the number of consumable additional items in the stock exceeds the fixed threshold. Stationary distributions of the system states and the waiting time are computed. In the numerical example, we show some interesting effects and illustrate a possibility of application of the presented results for solution of optimization problems.

Queues with semi-Markovian arrivals

1967 ◽  
Vol 4 (2) ◽  
pp. 365-379 ◽  
Author(s):  
Erhan Çinlar
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
Finite Number ◽  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
Interarrival Time ◽  
Single Server ◽  
Queue Size

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.

Loss Probability of a Burst Arrival Finite Queue with Synchronized Service

1992 ◽  
Vol 6 (2) ◽  
pp. 201-216 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Markov Chain ◽  
Loss Probability ◽  
Buffer Size ◽  
Embedded Markov Chain ◽  
Main Concern ◽  
Single Server ◽  
Finite Buffer ◽  
Single Server Queue ◽  
Server Queue ◽  
Computation Algorithm

We are concerned with a burst arrival single-server queue, where arrivals of cells in a burst are synchronized with a constant service time. The main concern is with the loss probability of cells for the queue with a finite buffer. We analyze an embedded Markov chain at departure instants of cells and get a kind of lumpability for its state space. Based on these results, this paper proposes a computation algorithm for its stationary distribution and the loss probability. Closed formulas are obtained for the first two moments of the numbers of cells and active bursts when the buffer size is infinite.

A Single Server Retrial Queueing System with Two Types of Batch Arrivals

2016 ◽  
pp. 55-70
Author(s):  
Kalyanaraman Rathinasabapathy
Keyword(s):  
Queueing System ◽  
Numerical Study ◽  
Priority Queue ◽  
Single Server ◽  
Type I ◽  
Type Ii ◽  
Operating Characteristics ◽  
Batch Arrivals ◽  
Retrial Queueing System ◽  
Retrial Queueing

A retrial queueing system with two types of batch arrivals is considered. The arrivals are called type I and type II customers. The type I customers arrive in batches of size k with probability c_k and type II customers arrive in batches of size k with probability d_k. Service time distributions are identical independent distributions and are different for both type of customers. If the arriving customers are blocked due to server being busy, type I customers are queued in a priority queue of infinity capacity whereas type II customers entered into retrial group in order to seek service again after a random amount of time. For this model the joint distribution of the number of customers in the priority queue and in the retrial group in closed form is obtained. Some particular models and operating characteristics are obtained. A numerical study is also carried out.

WAITING TIME ANALYSIS OF MULTI-CLASS QUEUES WITH IMPATIENT CUSTOMERS

2013 ◽  
Vol 27 (3) ◽  
pp. 333-352 ◽  
Author(s):  
Vahid Sarhangian ◽  
Bariş Balciog̃lu
Keyword(s):  
Waiting Time ◽  
Delay Systems ◽  
Single Server ◽  
Impatient Customers ◽  
Preemptive Priority ◽  
System A ◽  
Virtual Waiting Time ◽  
The Laplace Transform ◽  
Time Requirements ◽  
Priority Policy

In this paper, we study three delay systems where different classes of impatient customers arrive according to independent Poisson processes. In the first system, a single server receives two classes of customers with general service time requirements, and follows a non-preemptive priority policy in serving them. Both classes of customers abandon the system when their exponentially distributed patience limits expire. The second system comprises parallel and identical servers providing the same type of service for both classes of impatient customers under the non-preemptive priority policy. We assume exponential service times and consider two cases depending on the time-to-abandon distribution being exponentially distributed or deterministic. In either case, we permit different reneging rates or patience limits for each class. Finally, we consider the first-come-first-served policy in single- and multi-server settings. In all models, we obtain the Laplace transform of the virtual waiting time for each class by exploiting the level-crossing method. This enables us to compute the steady-state system performance measures.

On a property of a refusals stream

1997 ◽  
Vol 34 (03) ◽  
pp. 800-805 ◽  
Author(s):  
Vyacheslav M. Abramov
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
Elementary Proof ◽  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Wide Family

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

scholarly journals The Multiphase Queuing System of the Rijeka Airport

Pomorstvo ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 205-209
Author(s):  
Svjetlana Hess ◽  
Ana Grbčić
Keyword(s):  
Performance Indicators ◽  
Queueing Theory ◽  
Rare Case ◽  
Queueing System ◽  
Service System ◽  
Waiting Times ◽  
Real Problem ◽  
Single Server ◽  
Scientific Contribution ◽  
The Real

The paper gives an overview of the real system as a multiphase single server queuing problem, which is a rare case in papers dealing with the application of the queueing theory. The methodological and scientific contribution of this paper is primarily in setting up the model of the real problem applying the multiphase queueing theory. The research of service system at Rijeka Airport may allow the airport to be more competitive by increasing service quality. The existing performance measures have been evaluated in order to improve Rijeka Airport queueing system, as a record number of passengers is to be expected in the next few years. Performance indicators have pointed out how the system handles congestion. The research is also focused on defining potential bottlenecks and comparing the results with IATA guidelines in terms of maximum waiting times.

scholarly journals Concavity of the throughput of tandem queueing systems with finite buffer storage space

1990 ◽  
Vol 22 (03) ◽  
pp. 764-767 ◽  
Author(s):  
Ludolf E. Meester ◽  
J. George Shanthikumar
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Sample Path ◽  
Concave Function ◽  
Time Interval ◽  
Single Server ◽  
Finite Buffer ◽  
Buffer Storage ◽  
Unlimited Supply ◽  
Number Of Customers

We consider a tandem queueing system with m stages and finite intermediate buffer storage spaces. Each stage has a single server and the service times are independent and exponentially distributed. There is an unlimited supply of customers in front of the first stage. For this system we show that the number of customers departing from each of the m stages during the time interval [0, t] for any t ≧ 0 is strongly stochastically increasing and concave in the buffer storage capacities. Consequently the throughput of this tandem queueing system is an increasing and concave function of the buffer storage capacities. We establish this result using a sample path recursion for the departure processes from the m stages of the tandem queueing system, that may be of independent interest. The concavity of the throughput is used along with the reversibility property of tandem queues to obtain the optimal buffer space allocation that maximizes the throughput for a three-stage tandem queue.

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