Marked Markovian Arrival Process Research Articles

Two-types of service facilities in interconnected stochastic queueing and queueing-inventory system with marked Markovian arrival process

This article explores the performance of interconnected queueing systems and queueing-inventory systems (IQSQIS) in stochastic modeling. IQSQIS features two types of service facilities with dual service stations: one providing non-commodity service with a multi-server, and another handling commodity sales with a single server. As directed, a customer type i,i=1,2 can approach the station i,i=1,2 whose arrival pattern follows the marked Markovian arrival process (MMAP) since we assume that both service stations have an equal arrival phase. The IQSQIS gives an offer to the type-2 customer to choose type-1 service at the end of their service completion. Numerical results suggest that this option reduces customer wait times, orbit size, and overall system costs. The comparative analysis of heterogeneous and homogeneous servers in station-1 and queue-dependent and non-queue-dependent service facilities in station-2 is presented and investigated using the numerical outputs.

Optimization of traffic control in $ MMAP\mathit{[2]}/PH\mathit{[2]}/S$ priority queueing model with $ PH $ retrial times and the preemptive repeat policy

<p style='text-indent:20px;'>The presented study elaborates a multi-server priority queueing model considering the preemptive repeat policy and phase-type distribution (<inline-formula><tex-math id="M3">\begin{document}$ P\!H $\end{document}</tex-math></inline-formula>) for retrial process. The incoming heterogeneous calls are categorized as handoff calls and new calls. The arrival and service processes of both types of calls follow marked Markovian arrival process (<inline-formula><tex-math id="M4">\begin{document}$ M\!M\!A\!P $\end{document}</tex-math></inline-formula>) and <inline-formula><tex-math id="M5">\begin{document}$ P\!H $\end{document}</tex-math></inline-formula> distribution with distinct parameters, respectively. An arriving new call will be blocked when all the channels are occupied, and consequently will join the orbit (virtual space) to retry following <inline-formula><tex-math id="M6">\begin{document}$ P\!H $\end{document}</tex-math></inline-formula> distribution. When all the channels are occupied and a handoff call arrives at the system, out of the following two scenarios, one might take place. In the first scenario, if all the channels are occupied with handoff calls, the arriving handoff call will be lost from the system. While in the second one, if all the channels are occupied and at least one of them is serving a new call, the arriving handoff call will be provided service by using preemptive priority over that new call and the preempted new call will join the orbit. Behaviour of the proposed system is modelled by the level dependent quasi-birth-death <inline-formula><tex-math id="M7">\begin{document}$ (L\!D\!Q\!B\!D) $\end{document}</tex-math></inline-formula> process. The expressions of various performance measures have been derived for the numerical illustration. An optimization problem for optimal channel allocation and traffic control has been formulated and dealt by employing appropriate heuristic approaches.</p>

Performability analysis of a MMAP[2]/PH[2]/S model with PH retrial times

This work focuses over the performability analysis of a multi-server retrial queueing model with phase-type inter-retrial times in cellular networks. It is considered that the pattern of the new call arrival and handoff call arrival follows marked Markovian arrival process. The service times of both types of calls are phase-type (PH) distributed with different parameters, and inter-failure & inter-repair times of channels are exponentially distributed. For the prioritization of handoff calls, G channels are kept in reserve for handoff calls. When all the available channels, say S , are busy at the arrival epoch of a handoff call, the handoff call will be dropped. Whereas a new call will be blocked and will have an option to join the orbit of infinite capacity or leave the system without getting the connection, if at least S − G channels are busy. A new call in the orbit, termed as retrial call, retries to get the connection after a random interval which follows PH distribution. This model is analyzed as a level-dependent-quasi-birth-death process by applying an efficient method. Through numerical illustrations, the behavior of performance measures depending on the various relevant intensities is discussed.

Performance analysis of an energy-saving strategy in cloud data centers based on a MMAP[K]/M[K]/[formula omitted] non-preemptive priority queue

With the development of cloud computing, big data, artificial intelligence and other next generation information technologies, the scale of the data center industry worldwide is growing rapidly, resulting in a sharp increase in energy consumption. In order to efficiently reduce the idle energy consumption in cloud data centers, in this paper, we propose an energy-saving strategy based on two-threshold hysteresis cluster scheduling mechanism, which enables the reserved cluster to be dynamically switched on and off with load variation. In addition, the tasks to be processed are classified into real-time tasks and non-real-time tasks. To keep track of the two classes of tasks and model the correlated traffic in cloud data centers, we describe the arrival flow of tasks as a Marked Markovian Arrival Process (MMAP). Accordingly, we develop a non-preemptive priority queue as the system model to capture the working principle of the proposed strategy. By using the matrix-geometric solution and Gauss–Seidel method, the steady-state distribution of the system model is analyzed, and some key Quality of Service (QoS) metrics and Total Cost of Ownership (TCO) metrics are calculated. Results of numerical experiments show that, under the proposed energy-saving strategy, the overall power consumption in a small cloud data center can be reduced by an average of 30.20% under different scenarios, and the larger the cloud data center scale, the more obvious the energy saving effect. The results also confirm the impact of inter-class correlations of the input process on performance metrics. Furthermore, we identify the Pareto optimal solutions for trading off the overall power consumption and average waiting time of real-time tasks.

A retrial queueing system with processor sharing and impatient customers

Objectives. The problem of constructing and investigating a mathematical model of a stochastic system with processor sharing, repeated calls, and customer impatience is considered. This system is formalized in the form of a queueing system. The operation of the queue is described in terms of multi-dimensional Markov chain. A condition for the existence of a stationary distribution is found, and algorithms for calculating the stationary distribution and stationary performance characteristics of the system are proposed.Methods. Methods of probability theory, queueing theory and matrix theory are used.Results. The steady state operation of a queueing system with repeated calls, processor sharing and two types of customers arriving in a marked Markovian arrival process is studied. The channel bandwidth is divided between two types of customers in a certain proportion, and the number of customers of each type simultaneously located on the server is limited. Customers of one of the types that have made all the channels assigned to them busy leave the system unserved with some probability and, with an additional probability, go to the orbit of infinite size, from where they make attempts to get service at random time intervals. Customers of the second type, which caused all the channels assigned to them to be busy, are lost. Customers in orbit show impatience: each of them can leave orbit forever if the time of its stay in orbit exceeds some random time distributed according to an exponential law. Service times of customers of different types are distributed according to the phase law with different parameters. The operation of the system is described in terms of a multi-dimensional Markov chain. It is proved that for any values of the system parameters this chain has a stationary distribution. Algorithms for calculating the stationary distribution and a number of performance measures of the system are proposed. The results of the study can be used to simulate the operation of a fixed capacity cell in a wireless cellular communication network and other real systems operating in the processor sharing mode.

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

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.

Performance Evaluation of the Priority Multi-Server System MMAP/PH/M/N Using Machine Learning Methods

In this paper, we present the results of a study of a priority multi-server queuing system with heterogeneous customers arriving according to a marked Markovian arrival process (MMAP), phase-type service times (PH), and a queue with finite capacity. Priority traffic classes differ in PH distributions of the service time and the probability of joining the queue, which depends on the current length of the queue. If the queue is full, the customer does not enter the system. An analytical model has been developed and studied for a particular case of a queueing system with two priority classes. We present an algorithm for calculating stationary probabilities of the system state, loss probabilities, the average number of customers in the queue, and other performance characteristics for this particular case. For the general case with K priority classes, a new method for assessing the performance characteristics of complex priority systems has been developed, based on a combination of machine learning and simulation methods. We demonstrate the high efficiency of the new method by providing numerical examples.

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

In this paper, we consider two single server queueing systems to which customers of two distinct priorities (P1 and P2) arrive according to a Marked Markovian arrival process (MMAP). They are served according to two distinct phase type distributions. The probability of a P1 customer to feedback is θ on completion of his service. The feedback (P1) customers, as well as P2 customers, join the low priority queue. Low priority (P2) customers are taken for service from the head of the line whenever the P1 queue is found to be empty at the service completion epoch. We assume a finite waiting space for P1 customers and infinite waiting space for P2 customers. Two models are discussed in this paper. In model I, we assume that the service of P2 customers is according to a non-preemptive service discipline and in model II, the P2 customers service follow a preemptive policy. No feedback is permitted to customers in the P2 line. In the steady state these two models are compared through numerical experiments which reveal their respective performance characteristics.

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

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.

Priority Multi-Server Queueing System with Heterogeneous Customers

In this paper, we analyze a multi-server queueing system with heterogeneous customers that arrive according to a marked Markovian arrival process. Customers of two types differ in priorities and parameters of phase type distribution of their service time. The queue under consideration can be used to model the processes of information transmission in telecommunication networks in which often the flow of information is the superposition of several types of flows with correlation of inter-arrival times within each flow and cross-correlation. We define the process of information transmission as the multi-dimensional Markov chain, derive the generator of this chain and compute its stationary distribution. Expressions for computation of various performance measures of the system, including the probabilities of loss of customers of different types, are presented. Output flow from the system is characterized. The presented numerical results confirm the high importance of account of correlation in the arrival process. The values of important performance measures for the systems with the correlated arrival process are essentially different from the corresponding values for the systems with the stationary Poisson arrival process. Measurements in many real world systems show poor approximation of real flows by such an arrival process. However, this process is still popular among the telecommunication engineers due to the evident existing gap between the needs of adequately modeling the real life systems and the current state of the theory of algorithmic methods of queueing theory.

Mathematical Models for the Operation of a Cell With Bandwidth Sharing and Moving Users

A mathematical model is proposed for the operation of a cell in a mobile communication network with moving users. The cell is divided into several zones distinguished by the strength of the signal and, consequently, by the transmission rate. The process of users activation in various zones is defined by the marked Markovian arrival process. After activation, a user can move and transit to other zones or to another cell. The service rate of an arbitrary user depends on the zone where the user is currently located and the total number of active users in the cell. When the total bandwidth required by all active users exceeds the bandwidth of the base station, a proportional reduction of the bitrate provided to each user (bandwidth sharing) is applied. The key performance indicators of the system are computed, and illustrative numerical examples are presented.

Queueing Network with Moving Servers as a Model of Car Sharing Systems

We consider a queueing network with a finite number of nodes and servers moving between the nodes as a model of car sharing. The arrival process of customers to various nodes is defined by a marked Markovian arrival process. The customer that arrives at a certain node when there is no idle server (car) is lost. Otherwise, he/she is able to start the service. With known probability, which depends on the node and the number of available cars, this customer can balk the service and leave the system. The service time of a customer has an exponential distribution. Location of the server in the network after service completion is random with the known probability distribution. The behaviour of the network is described by a multi-dimensional continuous-time Markov chain. The generator of this chain is derived which allows us to compute the stationary distribution of the network states. The formulas for computing the key performance indicators of the system are given. Numerical results are presented. They characterize the dependence of some performance measures of the network and the nodes on the total number of cars (fleet size of the car sharing system) and correlation in the arrival process.

Analysis of a Semi-Open Queuing Network with a State Dependent Marked Markovian Arrival Process, Customers Retrials and Impatience

We consider a queuing network with single-server nodes and heterogeneous customers. The number of customers, which can obtain service simultaneously, is restricted. Customers that cannot be admitted to the network upon arrival make repeated attempts to obtain service. The service time at the nodes is exponentially distributed. After service completion at a node, the serviced customer can transit to another node or leave the network forever. The main features of the model are the mutual dependence of processes of customer arrivals and retrials and the impatience and non-persistence of customers. Dynamics of the network are described by a multidimensional Markov chain with infinite state space, state inhomogeneous behavior and special structure of the infinitesimal generator. The explicit form of the generator is derived. An effective algorithm for computing the stationary distribution of this chain is recommended. The expressions for computation of the key performance measures of the network are given. Numerical results illustrating the importance of the account of the mentioned features of the model are presented. The model can be useful for capacity planning, performance evaluation and optimization of various wireless telecommunication networks, transportation and manufacturing systems.

A two-priority single server retrial queue with additional items

In this paper, we study a priority queueing-inventory problem with two types of customers. Arrival of customers follows Marked Markovian arrival process and service times have phase-type distribution with parameters depending on the type of customer in service. For service of each type of customer, a certain number of additional items are needed. High priority customers do not have waiting space and so leave the system when on their arrival a priority 1 customer is in service or the number of available additional items is less than the required threshold. Preemptive priority is assumed. Type 2 customers, encountering a busy server or idle with the number of available additional items less than a threshold, go to an orbit of infinite capacity to retry for service. The customers in orbit are non-persistent: if on retrial the server is found to be busy/idle with the number of additional items less than the threshold, this customer abandons the system with certain probability. Such a system represents an accurate enough model of many real-world systems, including wireless sensor networks and system of cognitive radio with energy harvesting and healthcare systems. The probability distribution of the system states is computed, using which several of the characteristics are derived. A detailed numerical study of the system, including the analysis of the influence of the threshold, is performed.

A Queueing System with Heterogeneous Impatient Customers and Consumable Additional Items

AbstractA 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.

Novel Queueing Model for Multimedia Over Downlink in 3.5G Wireless Network

In this paper, a model for multimedia transmission over downlink shared channel in 3.5G wireless network is presented. The multimedia stream consists of multiplesubstreams that are aggregated into one real-time and onenonreal-time flows. Correlation with each flow and between flows is assumed. Additionally, we propose a combined time-space priority buffer management scheme to optimise quality of service requirements for each flow. The problem is formulated in terms of a queue with two priority classes, one of which has time priority while the another has space priority. The input is described by the Batch Marked Markovian Arrival Process (BMMAP). Service time distributions are of PH (phase) type dependent on the class of a customer. The buffer is finite, but the customers of a class having higher priority for taking into the service from a buffer (time priority) can occupy only a part of this buffer. Queueing system's behavior is described in terms ofmulti-dimensional continuous time skip-free to the left Markov chain. It allows to exploit an effective algorithm for calculation of the stationary distribution of the queueing system. Loss probability for customers of both classes is calculated. Waiting time distribution for priority customers is calculated.

Analysis of a Priority Queue with Phase-Type Service and Failures

We consider a single server queue with two types of customers. We propose a discipline of flexible priority in access that combines the features of randomization and the threshold type control. We introduce a new class of distributions, phase-type with failures (PHF) distribution, that generalizes the well-known phase-type (PH) distribution to the case when failures can occur during service of a customer. The arrival flow is described by the marked Markovian arrival process. The service time distribution is of PHF type with the parameters depending on the type of a customer. Customers of both types can be impatient. Behavior of the system is described by the multidimensional Markov chain. Problem of existence and computation of the stationary distribution of this Markov chain is discussed in brief as well as the problem of computation of the key performance measures of the system. Numerical examples are presented that give some insight into behavior of the system performance measures under different values of the parameters defining the strategy of customers access to service.

Hysteresis control by the number of active servers in queueing system [formula omitted] with priority service

The problem of choosing the optimal hysteresis strategy of control by the number of active servers in the multi-server queue is considered. Customers of two types arrive to the system according to the marked Markovian arrival process (MMAP). Type 1 customers have a non-preemptive priority, but the buffer for these customers is finite. The buffer for type 2 customers is infinite. The service time distribution is of phase-type (PH) depending on the type of customers. Some servers are always active. The rest of servers can be switched on or off depending on the number of customers in the system. The strategy of control by the number of active servers is of hysteresis type. Such a strategy is defined by two sets of thresholds. The servers are activated or switched off depending on the relation of the number of customers and the thresholds. The main contribution of the paper is development of a procedure for computation of the stationary distribution of the system states and the value of economical cost criterion under any fixed thresholds. Numerical results show effectiveness of the hysteresis control and importance of account of correlation in the arrival process and variance of service times.

Analysis and optimization of Guard Channel Policy with buffering in cellular mobile networks

A multi-server retrial queue with two types of calls (handover and new calls) is analyzed. This queue models the operation of a cell of a mobile communication network. Calls of two types arrive at the system according to the marked Markovian arrival process. The service times of both types of calls have phase-type distributions with different service rates. Handover calls have priority over new calls. Priority is ensured by provision of a dedicated buffer and reservation of several servers of the system exclusively for the service of handover calls. A handover call is dropped and leaves the system if all servers are busy and the buffer is full at the arrival epoch. During the waiting time in the buffer, the handover calls are impatient and may leave the system without service. A new call is blocked if all servers available to new calls are busy. Such a call has options to balk (to leave the system without obtaining service) or to retry later. The behavior of the system is described by the multi-dimensional Markov chain belonging to the class of the asymptotically quasi-Toeplitz Markov chains (AQTMC). In this paper, the generator of this Markov chain is constructed, and it is shown that the chain is ergodic for any set of the system parameters. Formulas for computing various performance measures of the system based on the stationary distribution of the Markov chain are derived. The results of numerical experiments illustrating the behavior of key performance measures of the system depending on the number of reserved servers are presented. A highly positive effect of buffering the handover calls and the server’s reservation is demonstrated.

Analysis of priority retrial queue with many types of customers and servers reservation as a model of cognitive radio system

Cognitive radio is emerging as one of the key information transmission technologies to enhance spectrum efficiency for dramatically increased wireless network capacity requested by end users. Dynamic spectrum access allows effective use of radio frequency and prevents its underutilization in many real-world networks. It enables unlicensed users to temporarily “borrow” unused spectrum while ensuring that the rights of the incumbent license holders are respected. Problems of optimization of joint access of primary and secondary users can be effectively solved by means of queueing theory. In this paper, the analysis of a novel queueing model suitable for the optimization of access is implemented under quite general assumptions about the system parameters. There are several types of primary customers having different requirements for the service time and preemptive priority over secondary customers. Secondary customers can share a server, while primary customers occupy the whole server. The arrival flow is described by the marked Markovian arrival process. The service time distribution is of phase-type. Effect of retrials of secondary customers is taken into account. An effective way for the analysis of multi-server queues with many types of customers and heterogeneous requirements to the service process is provided and applied.