Mathematics

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16 pages, 3406 KiB

Open AccessArticle

Mathematical and Simulation Model for Reliability Analysis of a Heterogeneous Redundant Data Transmission System

by Hector Gibson Kinmanhon Houankpo and Dmitry Kozyrev

Mathematics 2021, 9(22), 2884; https://doi.org/10.3390/math9222884 - 12 Nov 2021

Cited by 4 | Viewed by 1747

In the actual study, we carried out a reliability analysis of a repairable redundant data transmission system with the use of the elaborated mathematical and simulation model of a closed heterogeneous cold standby system. The system consists of one repair unit and two [...] Read more.

In the actual study, we carried out a reliability analysis of a repairable redundant data transmission system with the use of the elaborated mathematical and simulation model of a closed heterogeneous cold standby system. The system consists of one repair unit and two different data sources with an exponential cumulative distribution function (CDF) of their uptime and a general independent CDF of their repair time. We consider five special cases of the general independent CDF; including Gamma, Weibull-Gnedenko, Exponential, Lognormal and Pareto. We study the system-level reliability, defined as the steady-state probability (SSP) of failure-free system operation. The proposed analytical methodology made it possible to assess the reliability of the whole system in the event of failure of its components. Specific analytic expressions and asymptotic valuations are obtained for the steady-state probabilities of the system and the SSP of failure-free system operation. A simulation model of the system in cases where it is not workable to obtain expressions for the steady-state probabilities of the system in an explicit analytical form was considered, in particular for constructing the empirical system reliability function. The issue of sensitivity analysis of reliability characteristics of the considered system to the types of repair time distributions was also studied. The simulation modeling was done with the R statistics package. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)

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15 pages, 268 KiB

Open AccessArticle

The Domain of Residual Lifetime Attraction for the Classical Distributions of the Reliability Theory

by Vladimir Rusev and Alexander Skorikov

Mathematics 2021, 9(21), 2831; https://doi.org/10.3390/math9212831 - 08 Nov 2021

Viewed by 1544

Abstract

The asymptotic behavior of the residual lifetime of the system and its characteristics are studied for the main distributions of reliability theory. Sufficiently precise and simple conditions for the domain of attraction of the exponential distribution are proposed, which are applicable for a [...] Read more.

The asymptotic behavior of the residual lifetime of the system and its characteristics are studied for the main distributions of reliability theory. Sufficiently precise and simple conditions for the domain of attraction of the exponential distribution are proposed, which are applicable for a wide class of distributions. This approach allows us to take into account important information about modeling the failure-free operation of equipment that has worked reliably for a long time. An analysis of the domain of attraction for popular distributions with “heavy tails” is given. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)

18 pages, 730 KiB

Open AccessArticle

Optimal Open-Loop Routing and Threshold-Based Allocation in TWO Parallel QUEUEING Systems with Heterogeneous Servers

by Dmitry Efrosinin and Natalia Stepanova

Mathematics 2021, 9(21), 2766; https://doi.org/10.3390/math9212766 - 31 Oct 2021

Cited by 1 | Viewed by 1392

Abstract

In this paper, we study the problem of optimal routing for the pair of two-server heterogeneous queues operating in parallel and subsequent optimal allocation of customers between the servers in each queue. Heterogeneity implies different servers in terms of speed of service. An [...] Read more.

In this paper, we study the problem of optimal routing for the pair of two-server heterogeneous queues operating in parallel and subsequent optimal allocation of customers between the servers in each queue. Heterogeneity implies different servers in terms of speed of service. An open-loop control assumes the static resource allocation when a router has no information about the state of the system. We discuss here the algorithm to calculate the optimal routing policy based on specially constructed Markov-modulated Poisson processes. As an alternative static policy, we consider an optimal Bernoulli splitting which prescribes the optimal allocation probabilities. Then, we show that the optimal allocation policy between the servers within each queue is of threshold type with threshold levels depending on the queue length and phase of an arrival process. This dependence can be neglected by using a heuristic threshold policy. A number of illustrative examples show interesting properties of the systems operating under the introduced policies and their performance characteristics. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)

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20 pages, 3591 KiB

Open AccessArticle

New Importance Measures Based on Failure Probability in Global Sensitivity Analysis of Reliability

by Zdeněk Kala

Mathematics 2021, 9(19), 2425; https://doi.org/10.3390/math9192425 - 29 Sep 2021

Cited by 36 | Viewed by 7948

Abstract

This article presents new sensitivity measures in reliability-oriented global sensitivity analysis. The obtained results show that the contrast and the newly proposed sensitivity measures (entropy and two others) effectively describe the influence of input random variables on the probability of failure P_f [...] Read more.

This article presents new sensitivity measures in reliability-oriented global sensitivity analysis. The obtained results show that the contrast and the newly proposed sensitivity measures (entropy and two others) effectively describe the influence of input random variables on the probability of failure P_f. The contrast sensitivity measure builds on Sobol, using the variance of the binary outcome as either a success (0) or a failure (1). In Bernoulli distribution, variance P_f(1 − P_f) and discrete entropy—P_fln(P_f) − (1 − P_f)ln(1 − P_f) are similar to dome functions. By replacing the variance with discrete entropy, a new alternative sensitivity measure is obtained, and then two additional new alternative measures are derived. It is shown that the desired property of all the measures is a dome shape; the rise is not important. Although the decomposition of sensitivity indices with alternative measures is not proven, the case studies suggest a rationale structure of all the indices in the sensitivity analysis of small P_f. The sensitivity ranking of input variables based on the total indices is approximately the same, but the proportions of the first-order and the higher-order indices are very different. Discrete entropy gives significantly higher proportions of first-order sensitivity indices than the other sensitivity measures, presenting entropy as an interesting new sensitivity measure of engineering reliability. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)

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23 pages, 388 KiB

Open AccessArticle

Application of Decomposable Semi-Regenerative Processes to the Study of k-out-of-n Systems

by Vladimir Rykov, Nika Ivanova and Dmitry Kozyrev

Mathematics 2021, 9(16), 1933; https://doi.org/10.3390/math9161933 - 13 Aug 2021

Cited by 9 | Viewed by 1541

Abstract

This paper aimed to demonstrate the capabilities of decomposable semi-regenerative processes for the investigation of the k-out-of-n system. Proposed in 1955 by W. Smith, the regeneration idea has come a long way in terms of development and has found widespread applications. [...] Read more.

This paper aimed to demonstrate the capabilities of decomposable semi-regenerative processes for the investigation of the k-out-of-n system. Proposed in 1955 by W. Smith, the regeneration idea has come a long way in terms of development and has found widespread applications. First, we briefly recall the history of the development of the regeneration idea and the main results of the theory of regenerative, semi-regenerative, and decomposable semi-regenerative processes. Then, the methods of the theory of decomposable semi-regenerative processes are used for the study of a k-out-of-n renewable system with exponentially distributed life and generally distributed repair times of its components. This system is very important for practice and its special cases have previously been considered (including by the authors); however, only special cases and using other methods are considered herein. In the current paper, two scenarios of system repair after its failure are considered for the first time: the partial and the full system repair scenarios. For both scenarios, the time-dependent system state probabilities are calculated in terms of their Laplace transforms. The closed form representation of the stationary probabilities for both scenarios are also presented. These latest results represent a new contribution to the study of this system. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)

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26 pages, 388 KiB

Open AccessArticle

MMAP/(PH,PH)/1 Queue with Priority Loss through Feedback

by Divya Velayudhan Nair, Achyutha Krishnamoorthy, Agassi Melikov and Sevinj Aliyeva

Mathematics 2021, 9(15), 1797; https://doi.org/10.3390/math9151797 - 29 Jul 2021

Cited by 6 | Viewed by 1543

Abstract

In this paper, we consider two single server queueing systems to which customers of two distinct priorities (

P_{1}

and

P_{2}

) arrive according to a Marked Markovian arrival process (MMAP). They are served according to two distinct phase type distributions. [...] Read more.

In this paper, we consider two single server queueing systems to which customers of two distinct priorities (

P_{1}

and

P_{2}

) arrive according to a Marked Markovian arrival process (MMAP). They are served according to two distinct phase type distributions. The probability of a

P_{1}

customer to feedback is

θ

on completion of his service. The feedback (

P_{1}

) customers, as well as

P_{2}

customers, join the low priority queue. Low priority (

P_{2}

) customers are taken for service from the head of the line whenever the

P_{1}

queue is found to be empty at the service completion epoch. We assume a finite waiting space for

P_{1}

customers and infinite waiting space for

P_{2}

customers. Two models are discussed in this paper. In model I, we assume that the service of

P_{2}

customers is according to a non-preemptive service discipline and in model II, the

P_{2}

customers service follow a preemptive policy. No feedback is permitted to customers in the

P_{2}

line. In the steady state these two models are compared through numerical experiments which reveal their respective performance characteristics. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)

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23 pages, 446 KiB

Open AccessArticle

A Multi-Server Heterogeneous Queuing-Inventory System with Class-Dependent Inventory Access

by Karumbathil Rasmi, Machuveettil Joseph Jacob, Alexander S. Rumyantsev and Achyutha Krishnamoorthy

Mathematics 2021, 9(9), 1037; https://doi.org/10.3390/math9091037 - 03 May 2021

Cited by 7 | Viewed by 1725

Abstract

In this paper, we consider a queuing inventory system with heterogeneous customers of K types arriving according to a marked Markovian arrival process. Each class of customers differs by nature of the service they seek and different priorities are assigned for each class [...] Read more.

In this paper, we consider a queuing inventory system with heterogeneous customers of K types arriving according to a marked Markovian arrival process. Each class of customers differs by nature of the service they seek and different priorities are assigned for each class resulting in different levels of inventory admitted to exhaust for customers of each class. A single service node is provided for each class with exponential services having class-dependent service rates. All classes of customers are served from a single source of inventory replenished according to

(s, S)

policy with exponentially distributed lead time. Stability condition and steady state probabilities are obtained by matrix-analytic method. Some important performance measures are also derived. Inventory recycle time was analyzed in detail. Useful cost function and numerical illustrations are also given. The optimization problem is interesting and can be solved in similar real scenario. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)

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16 pages, 469 KiB

Open AccessArticle

Analysis of a k-Stage Bulk Service Queuing System with Accessible Batches for Service

by Achyutha Krishnamoorthy, Anu Nuthan Joshua and Vladimir Vishnevsky

Mathematics 2021, 9(5), 559; https://doi.org/10.3390/math9050559 - 06 Mar 2021

Cited by 9 | Viewed by 2079

Abstract

In most of the service systems considered so far in queuing theory, no fresh customer is admitted to a batch undergoing service when the number in the batch is less than a threshold. However, a few researchers considered the case of customers accessing [...] Read more.

In most of the service systems considered so far in queuing theory, no fresh customer is admitted to a batch undergoing service when the number in the batch is less than a threshold. However, a few researchers considered the case of customers accessing ongoing service batch, irrespective of how long service was provided to that batch. A queuing system with a different kind of accessibility that relates to a real situation is studied in the paper. Consider a single server queuing system in which the service process comprises of k stages. Customers can enter the system for service from a node at the beginning of any of these stages (provided the pre-determined maximum service batch size is not reached) but cannot leave the system after completion of service in any of the intermediate stages. The customer arrivals to the first node occur according to a Markovian Arrival Process

(M A P)

. An infinite waiting room is provided at this node. At all other nodes, with finite waiting rooms (waiting capacity

c_{j}, 2 \leq j \leq k

), customer arrivals occur according to distinct Poisson processes with rates

λ_{j}, 2 \leq j \leq k

. The service is provided according to a general bulk service rule, i.e., the service process is initiated only if at least a customers are present in the queue at node 1 and the maximum service batch size is b. Customers can join for service from any of the subsequent nodes, provided the number undergoing service is less than b. The service time distribution in each phase is exponential with service rate

μ_{j}^{m}

, which depends on the service stage

j, 1 \leq j \leq k

, and the size of the batch

m, a \leq m \leq b

. The behavior of the system in steady-state is analyzed and some important system characteristics are derived. A numerical example is presented to illustrate the applicability of the results obtained. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)

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29 pages, 1219 KiB

Open AccessArticle

Analysis of a Batch Arrival, Batch Service Queuing-Inventory System with Processing of Inventory While on Vacation

by Achyutha Krishnamoorthy, Anu Nuthan Joshua and Dmitry Kozyrev

Mathematics 2021, 9(4), 419; https://doi.org/10.3390/math9040419 - 20 Feb 2021

Cited by 18 | Viewed by 2574

Abstract

A single-server queuing-inventory system in which arrivals are governed by a batch Markovian arrival process and successive arrival batch sizes form a finite first-order Markov chain is considered in this paper. Service is provided in batches according to a batch Markovian service process, [...] Read more.

A single-server queuing-inventory system in which arrivals are governed by a batch Markovian arrival process and successive arrival batch sizes form a finite first-order Markov chain is considered in this paper. Service is provided in batches according to a batch Markovian service process, with consecutive service batch sizes forming a finite first-order Markov chain. A service starts for the next batch on completion of the current service, provided that inventory is available at that epoch; otherwise, there will be a delay in starting the next service. When the service of a batch is completed, the inventory decreases by 1 unit, irrespective of batch size. A control policy in which the server goes on vacation when a service process is frozen until a quorum can initiate the next batch service is proposed to ensure idle-time utilization. During the vacation, the server produces inventory (items) for future services until it hits a specified level L or until the number of customers in the system reaches a maximum service batch size N, with whichever occurring first. In the former case, a server stays idle once the processed inventory level reaches L until the number of customers reaches (or even exceeds because of batch arrival) a maximum service batch size N. The time required for processing one unit of inventory follows a phase-type distribution. In this paper, the steady-state probability vector of this infinite system is computed. The distributions of inventory processing time in a vacation cycle, idle time in a vacation cycle, and vacation cycle length are found. The effect of correlation in successive inter-arrival times and service times on performance measures for such a queuing system is illustrated with a numerical example. An optimization problem is considered. The proposed system is then compared with a queuing-inventory system without the Markov-dependent assumption on successive arrivals as well as service batch sizes using numerical examples. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)

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27 pages, 354 KiB

Open AccessEditor’s ChoiceArticle

Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

by P.-C.G. Vassiliou

Mathematics 2021, 9(1), 55; https://doi.org/10.3390/math9010055 - 29 Dec 2020

Cited by 7 | Viewed by 1826

Abstract

For a

G

-inhomogeneous semi-Markov chain and

G

-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up [...] Read more.

For a

G

-inhomogeneous semi-Markov chain and

G

-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a

G

-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the

G

-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)

16 pages, 938 KiB

Open AccessArticle

Analysis of Instantaneous Feedback Queue with Heterogeneous Servers

by Agassi Melikov, Sevinj Aliyeva and János Sztrik

Mathematics 2020, 8(12), 2186; https://doi.org/10.3390/math8122186 - 08 Dec 2020

Cited by 6 | Viewed by 1480

Abstract

A system with heterogeneous servers, Markov Modulated Poisson flow and instantaneous feedback is studied. The primary call is serviced on a high-speed server, and after it is serviced, each call, according to the Bernoulli scheme, either leaves the system or requires re-servicing. After [...] Read more.

A system with heterogeneous servers, Markov Modulated Poisson flow and instantaneous feedback is studied. The primary call is serviced on a high-speed server, and after it is serviced, each call, according to the Bernoulli scheme, either leaves the system or requires re-servicing. After the completion of servicing of a call in a slow server, according to the Bernoulli scheme, it also either leaves the system or requires re-servicing. If upon arrival of a primary call the queue length of such calls exceeds a certain threshold value and the slow server is free, then the incoming primary call, according to the Bernoulli scheme, is either sent to the slow server or joins its own queue. A mathematical model of the studied system is constructed in the form of a three-dimensional Markov chain. Approximate algorithms for calculating the steady-state probabilities of the models with finite and infinite queues are proposed and their high accuracy is shown. The results of numerical experiments are presented. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)

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4 pages, 224 KiB

Open AccessArticle

Distributional Properties of Fluid Queues Busy Period and First Passage Times

by Zbigniew Palmowski

Mathematics 2020, 8(11), 1988; https://doi.org/10.3390/math8111988 - 07 Nov 2020

Cited by 1 | Viewed by 1224

Abstract

In this paper, I analyze the distributional properties of the busy period in an on-off fluid queue and the first passage time in a fluid queue driven by a finite state Markov process. In particular, I show that the first passage time has [...] Read more.

In this paper, I analyze the distributional properties of the busy period in an on-off fluid queue and the first passage time in a fluid queue driven by a finite state Markov process. In particular, I show that the first passage time has a IFR distribution and the busy period in the Anick-Mitra-Sondhi model has a DFR distribution. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)

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