Research

16 pages, 512 KiB

Open AccessArticle

Single-Server Queuing-Inventory Systems with Negative Customers and Catastrophes in the Warehouse

by Agassi Melikov, Laman Poladova, Sandhya Edayapurath and Janos Sztrik

Mathematics 2023, 11(10), 2380; https://0-doi-org.brum.beds.ac.uk/10.3390/math11102380 - 19 May 2023

Cited by 4 | Viewed by 1092

In this paper, we studied single-server models of queuing-inventory systems (QIS) with catastrophes in the warehouse part and negative customers (n-customers) in service facility. Consumer customers (c-customers) that arrived to buy inventory can be queued in an infinite buffer. [...] Read more.

In this paper, we studied single-server models of queuing-inventory systems (QIS) with catastrophes in the warehouse part and negative customers (n-customers) in service facility. Consumer customers (c-customers) that arrived to buy inventory can be queued in an infinite buffer. Under catastrophes, all inventory of the system is destroyed but customers in the system (on server or in buffer) are still waiting for replenishment of stocks. Upon arrival of n-customer one c-customer is pushed out, if any. One of two replenishment policies (RP) can be used in the system: either

(s, S)

or randomized. In the investigated QISs, a hybrid service scheme was used: if upon arrival of the c-customer, the inventory level is zero, then according to the Bernoulli scheme, this customer is either lost (lost sale scheme) or joining the queue (backorder scheme). Mathematical models of the investigated QISs were constructed as two-dimensional Markov chains (2D MC). Ergodicity conditions of the investigated QISs were obtained, and the matrix-analytic method (MAM) was used to calculate the steady-state probabilities of the constructed 2D MCs. Formulas for performance measures were found and the results of numerical experiments are presented. Full article

(This article belongs to the Special Issue Advances in Queueing Theory)

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11 pages, 474 KiB

Open AccessArticle

Estimating the Rate of Convergence of the PH/M/1 Model by Reducing to Quasi-Birth-Death Processes

by Ilya Usov, Yacov Satin and Alexander Zeifman

Mathematics 2023, 11(6), 1494; https://0-doi-org.brum.beds.ac.uk/10.3390/math11061494 - 18 Mar 2023

Viewed by 851

Abstract

We are studying the quasi-birth-death process and the property of weak ergodicity. Using the C-matrix method, we derive estimates for the rate of convergence to the limiting regime for the general case of the

P H / M / 1

model, as well [...] Read more.

We are studying the quasi-birth-death process and the property of weak ergodicity. Using the C-matrix method, we derive estimates for the rate of convergence to the limiting regime for the general case of the

P H / M / 1

model, as well as the particular case when

m = 3

. We provide a numerical example for the case

m = 3

, and construct graphs showing the probability of an empty queue and the probability of

p_{1} (t)

. Full article

(This article belongs to the Special Issue Advances in Queueing Theory)

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21 pages, 767 KiB

Open AccessArticle

Analysis of a Queuing System with Possibility of Waiting Customers Jockeying between Two Groups of Servers

by Sergei A. Dudin, Olga S. Dudina and Olga I. Kostyukova

Mathematics 2023, 11(6), 1475; https://0-doi-org.brum.beds.ac.uk/10.3390/math11061475 - 17 Mar 2023

Cited by 3 | Viewed by 2140

Abstract

In this paper, we consider a queueing system consisting of two multi-server subsystems that is designed for the service of clients arriving at a system according to a Markovian arrival process (MAP). Arriving clients receive information about the number of clients [...] Read more.

In this paper, we consider a queueing system consisting of two multi-server subsystems that is designed for the service of clients arriving at a system according to a Markovian arrival process (MAP). Arriving clients receive information about the number of clients present in both subsystems and use this information to make a randomized decision to balk (depart without receiving service) or join the system. In the latter case, they also decide which subsystem they would like to join. One subsystem has an infinite buffer, while the buffer of the second subsystem is finite. The service time distribution is exponential in the first subsystem and phase-type in the second subsystem. During the waiting in the chosen buffers, after the random time intervals, each waiting client checks the status of the alternative subsystem. If some server in that subsystem is idle during this epoch, the client immediately leaves the buffer where it has been staying and starts a service in the alternative subsystem. The problem of computing the steady-state distribution of this system is solved. The feasibility of the proposed solution and certain features of the system’s behavior are numerically illustrated. Full article

(This article belongs to the Special Issue Advances in Queueing Theory)

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22 pages, 980 KiB

Open AccessArticle

Analysis of Multi-Server Queueing System with Flexible Priorities

by Konstantin Samouylov, Olga Dudina and Alexander Dudin

Mathematics 2023, 11(4), 1040; https://0-doi-org.brum.beds.ac.uk/10.3390/math11041040 - 18 Feb 2023

Cited by 2 | Viewed by 2486

Abstract

In this paper, a multi-server queueing system providing service to two correlated flows of requests was considered. Non-preemptive priority was granted to one flow via the preliminary delay of requests in the intermediate buffers with different rates of extracting from the buffers. Customers’ [...] Read more.

In this paper, a multi-server queueing system providing service to two correlated flows of requests was considered. Non-preemptive priority was granted to one flow via the preliminary delay of requests in the intermediate buffers with different rates of extracting from the buffers. Customers’ impatience during waiting in the intermediate and main buffers was taken into account. The possibility of the use of the results of the mathematical analysis for managerial goals is numerically illustrated. Full article

(This article belongs to the Special Issue Advances in Queueing Theory)

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13 pages, 868 KiB

Open AccessFeature PaperArticle

Optimization of Open Queuing Networks with Batch Services

by Elena Stankevich, Igor Tananko and Michele Pagano

Mathematics 2022, 10(16), 3027; https://0-doi-org.brum.beds.ac.uk/10.3390/math10163027 - 22 Aug 2022

Cited by 2 | Viewed by 1446

Abstract

In this paper, open queuing networks with Poisson arrivals and single-server infinite buffer queues are considered. Unlike traditional queuing models, customers are served (with exponential service time) in batches, so that the nodes are non-work-conserving. The main contribution of this work is the [...] Read more.

In this paper, open queuing networks with Poisson arrivals and single-server infinite buffer queues are considered. Unlike traditional queuing models, customers are served (with exponential service time) in batches, so that the nodes are non-work-conserving. The main contribution of this work is the design of an efficient algorithm to find the batch sizes which minimize the average response time of the network. As preliminary steps at the basis of the proposed algorithm, an analytical expression of the average sojourn time in each node is derived, and it is shown that this function, depending on the batch size, has a single minimum. The goodness of the proposed algorithm and analytical formula were verified through a discrete-event simulation for an open network with a non-tree structure. Full article

(This article belongs to the Special Issue Advances in Queueing Theory)

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13 pages, 329 KiB

Open AccessArticle

Tail Asymptotics for a Retrial Queue with Bernoulli Schedule

by Bin Liu and Yiqiang Q. Zhao

Mathematics 2022, 10(15), 2799; https://doi.org/10.3390/math10152799 - 07 Aug 2022

Cited by 3 | Viewed by 1012

Abstract

In this paper, we study the asymptotic behaviour of the tail probability of the number of customers in the steady-state

M / G / 1

retrial queue with Bernoulli schedule, under the assumption that the service time distribution has a regularly varying tail. [...] Read more.

In this paper, we study the asymptotic behaviour of the tail probability of the number of customers in the steady-state

M / G / 1

retrial queue with Bernoulli schedule, under the assumption that the service time distribution has a regularly varying tail. Detailed tail asymptotic properties are obtained for the conditional probability of the number of customers in the (priority) queue and orbit, respectively, in terms of the recently proposed exhaustive stochastic decomposition approach. Numerical examples are presented to show the impacts of system parameters on the tail asymptotic probabilities. Full article

(This article belongs to the Special Issue Advances in Queueing Theory)

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12 pages, 289 KiB

Open AccessArticle

Pseudo Steady-State Period in Non-Stationary Infinite-Server Queue with State Dependent Arrival Intensity

by Anatoly Nazarov, Alexander Dudin and Alexander Moiseev

Mathematics 2022, 10(15), 2661; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152661 - 28 Jul 2022

Cited by 5 | Viewed by 1144

Abstract

An infinite-server queueing model with state-dependent arrival process and exponential distribution of service time is analyzed. It is assumed that the difference between the value of the arrival rate and total service rate becomes positive starting from a certain value of the number [...] Read more.

An infinite-server queueing model with state-dependent arrival process and exponential distribution of service time is analyzed. It is assumed that the difference between the value of the arrival rate and total service rate becomes positive starting from a certain value of the number of customers in the system. In this paper, time until reaching this value by the number of customers in the system is called the pseudo steady-state period (

P S S P

). Distribution of duration of

P S S P,

its raw moments and its simple approximation under a certain scaling of the number of customers in the system are analyzed. Novelty of the considered problem consists of an arbitrary dependence of the rate of customer arrival on the current number of customers in the system and analysis of time until reaching from below a certain level by the number of customers in the system. The relevant existing papers focus on the analysis of time interval since exceeding a certain level until the number of customers goes down to this level (congestion period). Our main contribution consists of the derivation of a simple approximation of the considered time distribution by the exponential distribution. Numerical examples are presented, which confirm good quality of the proposed approximation. Full article

(This article belongs to the Special Issue Advances in Queueing Theory)

16 pages, 662 KiB

Open AccessArticle

Double Sources Queuing-Inventory System with Hybrid Replenishment Policy

by Agassi Melikov, Ramil Mirzayev and Sajeev S. Nair

Mathematics 2022, 10(14), 2423; https://0-doi-org.brum.beds.ac.uk/10.3390/math10142423 - 11 Jul 2022

Cited by 8 | Viewed by 1212

Abstract

A hybrid replenishment policy in double sources queuing-inventory system is proposed. If the inventory level drops to the reorder point s, then a regular order of the fixed volume Q = S − s is generated to a slow and cheap source, where [...] Read more.

A hybrid replenishment policy in double sources queuing-inventory system is proposed. If the inventory level drops to the reorder point s, then a regular order of the fixed volume Q = S − s is generated to a slow and cheap source, where S denotes the maximum size of the system’s warehouse. If the inventory level falls below a certain threshold value r, where r < s, then the system instantly cancels the regular order and generates an emergency order to a fast and expensive source where the replenishment quantity should be able to bring the inventory level back to S at the replenishment epoch. In addition to consuming customers, the system also receives destructive customers that do not require inventory but destroy them. The stability condition for the system under study is found, steady-state probabilities are calculated, and formulas for finding performance measures are proposed. The problem of minimizing the total cost of the system under the proposed hybrid replenishment policy is solved by choosing the appropriate values of the order point and the threshold value. Full article

(This article belongs to the Special Issue Advances in Queueing Theory)

11 pages, 547 KiB

Open AccessArticle

A Multi-Type Queueing Inventory System—A Model for Selection and Allocation of Spectra

by Thulaseedharan Salini Sinu Lal, Varghese Chaukayil Joshua, Vladimir Vishnevsky, Dmitry Kozyrev and Achyutha Krishnamoorthy

Mathematics 2022, 10(5), 714; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050714 - 24 Feb 2022

Cited by 3 | Viewed by 1613

Abstract

The model discussed in this paper provides an efficient mechanism for the selection and allocation of available limited spectra for transmission of heterogeneous data in a network. The data packets (customers), belonging to different classes, arrive according to a batch marked the Markovian [...] Read more.

The model discussed in this paper provides an efficient mechanism for the selection and allocation of available limited spectra for transmission of heterogeneous data in a network. The data packets (customers), belonging to different classes, arrive according to a batch marked the Markovian arrival process (BMMAP). The inventory considered is of multi-type (different types of channels becoming available) and are generated according to a marked Markovian arrival process (MMAP). The number of distinct types of inventory and that of the customers are the same. Arriving customers are allowed to wait in finite buffers of each category which are reserved for distinct classes of customers except for the most general class, which is provided with an infinite waiting space. The number of servers also equals the number of distinct types of inventory. When items of a particular type arrive in the inventory, the service starts, providing the buffer of customers of the corresponding class is non-empty. The service can be viewed as a selection process with Coxian distributed service times. The system is analyzed using the matrix analytic method and performance measures are obtained. The model is illustrated with suitable numerical examples. Full article

(This article belongs to the Special Issue Advances in Queueing Theory)

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