Information Technologies and Mathematical Modelling - Queueing Theory and Applications

In Chakravarthy [5] a new class of queueing models in which one type of customers opt for cooperative services with fellow customers was introduced in the context of a single server. Under the assumption of versatile Markovian point process for the arrivals, exponential services, and with a limit of no more than two groups of cooperative customers be present in the system, the model was analyzed in steady-state and some interesting numerical examples were illustrated in [5]. In this paper we generalize that cooperative services model by relaxing the assumptions of single server, exponential services, and only two groups be present at any given time. Thus, we consider a multi-server queueing model in which the customers arrive according to a versatile Markovian point process. One type of customers require individual services whereas the second type of customers opt for a cooperative service (to be offered along with other similar customers). We assume that at any given time there can be at most $$K, 2 < K \le \infty $$K,2<K≤∞, groups of customers needing cooperative services and that the services are of phase type with representation depending on the type of service offered. While this model can be analyzed using matrix-analytic method with a very large state space, in this paper we will study the model using simulation to bring out a few salient features of this new class of queueing models.

This paper is focused on studying the modulated semi-synchronous integrated flow of events which is one of the mathematical models for incoming streams of events (claims) in computer communication networks and is related to the class of doubly stochastic Poisson processes (DSPPs). The flow is considered in conditions of its incomplete observability, when the dead time period of a constant duration T is generated after every registered event. In this paper we propose a technique for obtaining the formulas for calculation the probability density of the interval length between two neighboring flow events and the joint probability density of the length of two successive intervals. Also we find the conditions of the flow recurrence.

In this paper, we provide a simple framework for applying the mean-field theory to dealing with a heterogeneous work stealing model of M clusters, each of which consists of N same servers and operates under two types of work stealing schemes: One within a cluster, and another between any two clusters. We first set up an infinite-dimensional system of mean-field equations, which is related to the M clusters. Then we use the martingale limit theory to prove the asymptotic independence of this heterogeneous work stealing model. Finally, we analyze and compute the fixed point, which can give performance analysis of this heterogeneous stealing model.

A modulated synchronous doubly stochastic flow under conditions of a fixed dead time is considered. After each registered event there is a time of fixed duration T (dead time), during which another flow events are inaccessible for observation. When duration of the dead time period finishes, the first happened event creates the dead time period of duration T again and etc. An explicit form of a probability density function of interval duration between two adjacent events of modulated synchronous doubly stochastic flow under conditions of a fixed dead time is derived. Also an explicit form of a joint probability density function for modulated synchronous flow interval duration is obtained. A recurrent conditions for modulated synchronous flow as well as some probabilistic characteristics of the flow are obtained using the formula for a joint probability density function.

Stationary functioning of a queueing network with batch negative customer arrivals is analyzed. Necessary and sufficient condition for ergodisity of the isolated node is established. Stationary product-form distribution of network states is found. Given network model is generalization of classic G-network model on the case of several types of negative customers.

This paper deals with a finite source retrial queueing system of type M/M/1//N with collision of the customers. This means that the system has one server and N sources. Analysis of the sojourn time in the system is presented. The analysis is performed under an asymptotic condition of infinitely increasing number of sources. The approximation of the distribution of the total sojourn time in the system is derived.

We consider the infinite-server queueing network with semi-Markov arrivals. The system of differential equations for characteristic function of customers number at the network nodes is derived. The system is solved under asymptotic condition of high-rate arrivals. It is shown that probability distribution of customers at the network nodes can be approximated by multi-dimensional Gaussian distribution which parameters are obtained in the paper. Presented results of numerical experiments allow to determine the approximation applicability.

The main objective of the paper is to investigate the distribution of the number of lost calls, made during the busy period in one single server finite queueing system. It is assumed that the customers, failed to get service are temporarily blocked in the orbit of inactive customers for an exponentially distributed time interval. This model and its variants have many applications, especially for optimization of the corresponding models with retrials. Using the discrete transformations method we derive formulas for computing the mean value of the number of lost calls made during the busy period.

A mathematical model of access method “carrier sense multiple access with collision avoidance” for two active stations was proposed. The effect of carrier capture and unimodal dependence of the operating characteristics from the initial width of the contention window was detected. Measures of preventing the effect of carrier capture, based on the modifications of the standard protocol were proposed. For research into the capture effect with a large number of rivals, a simulation model of the competition process was developed. The efficiency of prevention measures ensuring fair distribution of a jointly used time resource within a shared communication medium with an insignificant decrease in the general throughput of the access method has been shown.

We investigate tandem queueing systems with control of conflicting input flows using cyclic algorithms with readjustments. Input flows are modulated by a finite-state synchronous Markov chain. Customers arrive in Poisson flows of batches with intensities and batch size distributions determined by the environment. All serviced customers from the first input flow and randomly selected serviced customers from the second conflicting input flow in the first system are transferred with random speeds to the second queueing system. We develop a numerical algorithm to evaluate the stationary probability distribution for the number of customers joining the transfer queue at each stage of servers’ operation.

In this paper we consider an $$M/GI/\infty $$M/GI/∞ queueing system operating in a semi-Markovian random environment. That is, the arrival rate and service-time distribution change according to the external semi-Markov process state transitions. The service policy subject to environment transitions is as follows: the service-time distribution of the present customers does not change until their service is finished. The purpose of our study is to obtain the probability distribution of the number of customers in the system under asymptotic condition of high arrival rate and frequent environment transitions. To do this, we first apply the method of supplementary variable and the original method of dynamic screening to our system. We then conduct the asymptotic analysis of the system to obtain the discrete probability distribution.

We consider a modulated MAP flow of events with two states; it is one of the mathematical models for an incoming stream of claims (events) in digital integral servicing networks. The observation conditions for this flow are such that each event generates a period of dead time during which other events from the flow are inaccessible for observation and do not extend the dead time period (unextendable dead time). We find an explicit form for a probability density of interval duration between neighboring events in the observed flow. We consider the stationary operation mode for the observed flow, so we disregard the transient processes. The duration estimation of unextendable dead time is essential to determine the number of the lost events which have the useful information.

The approach originally developed for the investigation of the traffic, that is, the intensities of information flows in financial markets, is applied for the statistical analysis of climatic data. The statistical regularities in the behavior of sensible and latent turbulent heat fluxes recomputed from 6-hourly NCEP-NCAR for the period $$1948-2008$$1948-2008 in Atlantic are analyzed. It is proposed to represent these regularities by probability distributions that are mixtures of several normal (Gaussian) laws with parameters varying in time. The method of moving separation of mixtures is used to obtain the values of the parameters of the mixtures. This approach allows to analyze the regularities in the variation of the parameters and, hence, to capture the low-term variability which can be considered as a trend and high-term dynamics associated with diffusion or irregular variability.

This paper is devoted to the research of the model of insurance company with an unlimited insurance field and the parameter of arrival process of insurance risks, which depends on the risks that are already insured in the company. Using method of characteristic functions we got joint probability distribution of a two-dimensional stochastic process of a number of risks that are insured in the company and a number of benefit payments. We also got expressions for the expected values and variances of components of a two-dimensional process. Total benefit payments is reviewed and its distribution and numerical characteristic are found.

In this paper, we study the queuing system with unlimited queue and with N servers. We obtain the approximation of probability distribution of the number of customers in the system. We obtain the formula of the probability of immediate service and the characteristic function of a positive waiting time. The optimal number of servers can be determined by the obtained characteristics.

The new Markov models of multi-channel queueing systems with instantaneous and delayed feedback are proposed. In these models part of already serviced calls instantaneously feeds back to channel while the rest part either leaves the system or feeds back to channel after some delay in orbit. Behavior of already serviced calls is handled by randomized parameters. Both exact and asymptotic methods to calculate the quality of service (QoS) metrics of the proposed models are developed. Exact method is based on the system of balance equations (SBE) for steady-state probabilities of appropriate three dimensional Markov chain (3-D MC) while asymptotic method uses the new hierarchical space merging algorithm for 3-D MC. Results of numerical experiments are demonstrated.

In this paper, we study the retrial queueing system with two arrival processes and two orbits with r-persistent exclusion of alternative customers by method of asymptotic analysis under condition of long delay. Stationary probability distribution of server states and values of asymptotic means of the number of customers in the orbits are obtained.

We investigate queueing systems with demands having some random space requirements (capacities) and service times generally depending on their capacities. For such systems, we discus the problem of steady-state loss characteristics determination, calculate these characteristics for some special cases and present a way for their estimation.

The research of the queuing system with renewal arrival process, infinite number of n different types servers and arbitrary service time distribution is proposed. Expressions for the characteristic function of the number of busy servers for different types of customers in the system under the asymptotic condition that service time infinitely grows equivalently to each type of customers are derived.

We consider an unreliable single-server queueing system with a reserve (back-up) server which is appropriate for modeling, e.g., the hybrid communication systems having the atmospheric optic channel (FSO-Free Space Optics) and standby radio channel IEEE 802.11n. We assume that the main server of the system (optic channel, server 1) is unreliable while the reserve server (radio channel, server 2) is absolutely reliable. It is assumed that the service time on the server 1 is essentially smaller than the service time on the server 2. If the server 1 fails during the service of a customer, the customer immediately occupies the reserve server. After failure occurence, the server 1 immediately goes to repair. When the repair period of the server 1 ends and the customer is served by the server 2, the customer immediately moves to the server 1. We consider exponential distributions of all random variables describing the operation of the system. This assumption allows to minimize the use of matrix-geometric technique and provide more or less simple analytical analysis of the system. We derive ergodicity condition and determine the stationary distribution of the Markov chain describing the process of the system states using the generating functions and roots methods, calculate some key performance measures and derive an expression for the Laplace-Stieltjes transform of the sojourn time distribution of an arbitrary customer. Illustrative numerical results are presented.

We consider exact probability distributions of demand and selling time for compound Poisson demand process with exponential batch size’s distribution in the newsvendor problem’s framework and compare the results with the normal and diffusion approximation respectively. Also we receive the inventory level’s distribution for on-off control of inventory level when the process of delivering the product to outlets is also a compound Poisson independent from the first one.

We study an M / G / 1 queue with multiple vacation and vacation interruption. Both normal vacation (type I) and working vacation (type II) are considered. The exhaustive service discipline is assumed in this paper. At the end of a busy period, depending on the environment, the server either opts for normal vacation or working vacation. On completion of type I vacation if the server finds the system empty he goes for type II vacation. On completion of type II vacation if the server finds the system empty goes for another type II vacation. On completion of service in type II vacation, if the server finds one or more customers in queue he returns to normal service, interrupting the vacation. An arriving customer, during type I vacation, joins the queue with probability q or leaves the system with probability $$\displaystyle 1-q$$1-q and during type II vacation all the arriving customers join the queue. Using supplementary variable technique we derive the distributions for the queue length and service status under steady state condition. Laplace-Stieltjes transform of the stationary waiting time is also developed. Some numerical illustrations are also given.

In this paper we obtain the probability density function of stock of perishable goods under constant production and hysteresis control of the selling price.

The problem of Markov-modulated Poisson process intensities estimating is studied. A new approach based on sequential change point detection method is proposed to determine switching points of the flow parameter. Both the intensities of the controlling Markovian chain and the intensities of the flow of events are estimated. The results of simulation are presented.

The paper discusses various models of self-similar Internet traffic and techniques for estimating the intensity of Long-Range Dependence (LRD). In the experimental part real data sets collected in IITiS PAN are used together with synthetic LRD flows generated using Fractional Gaussian noise and Markov modulated Poisson processes. We are especially interested in Markov models since they can be incorporated in Markov queueing models, for which powerful analytical and numerical techniques are available.

European exotic put option with payment limitation for issuer and guaranteed income for holder of the security is researched when base risk active is share index. The equitable option price, the securities portfolio structure and a size of the capital answered the hedging strategy are founded for the option under consideration on diffusion (B, S)-financial market. Comparative price analysis for two option classes is carried out and specific properties of decision and decision under limiting are explored.

The model of retail outlet in form of queueing system of M/M/1/$$\infty $$∞ type with request rejection is proposed. The output flow of the system and the rejected request flow are researched. Average number of events occurred in these flows is determined. In conditions of increasing observation time the asymptotic distributions of probabilities of number of events that occurred in studied flows are found by means of asymptotic analysis.

In this paper we study a single-server Markovian retrial queueing system with non-reliable server and threshold-based recovery policy. The arrived customer finding a free server either gets service immediately or joins a retrial queue. The customer at the head of the retrial queue is allowed to retry for service. When the server is busy, it is subject to breakdowns. In a failed state the server can be repaired with respect to the threshold policy: the repair starts when the number of customers in the system reaches a fixed threshold level. Using a matrix-analytic approach we perform a stationary analysis of the system. The optimization problem with respect to the average cost criterion is studied. We derive expressions for the Laplace transforms of the waiting time. The problem of estimation and confidence interval construction for the fully observable system is studied as well.

In the paper, the retrial queueing system of MMPP|M|1 type is studied by means of the second order asymptotic analysis method under heavy load condition. During the investigation, the theorem about the form of the asymptotic characteristic function of the number of calls in the orbit is formulated and proved. The asymptotic distribution is compared with the exact one obtained by means of numerical algorithm. The conclusion about method application area is made.

The main signaling protocol for IP Multimedia Subsystem is Session Initiation Protocol. A typical procedure for a session establishment involves two types of signaling messages with different priority: Invite message, which initiate a session, and non-Invite messages, which continue session establishment. In the paper for the analysis of two types signaling messages service process the single server asymmetric polling system with two infinite queues and a non-zero switching time is proposed. We estimate the waiting time for the gated and the exhaustive service disciplines using input data applicable to signaling traffic analysis. For the exhaustive discipline the formulas for the second moments of the waiting time were obtained and calculation of the first and the second moments of the waiting time was carried out.

Stationary functioning of a closed queueing network with temporarily non-active customers and multi-regime service strategies is analyzed. Non-active customers are located in the network nodes in queues, being not serviced. For a customer, the opportunity of passing from its ordinary state to the temporarily non-active state (and backwards) is provided. Quantity of work for customer service is a random distributed value. Stationary distribution insensitivity with respect to functional form of distribution of work quantity for customer service is established.

In the paper, the dynamic and adaptive RQ-systems with incoming MAP-flow of requests are investigated with the method of asymptotic analysis. It is shown that the dynamic and the adaptive RQ-systems are asymptotically equivalent. The results obtained by investigating the dynamic RQ-systems can be used to determine the probability distribution of customers in the orbit of the adaptive RQ-systems.

According to analytics, the global mobile date traffic will grow three times faster than fixed traffic by 2019. The number of user’s mobile devices is supposed to increase from 4.1 billion to 4.9 billion while the number of mobile device connections can reach even 10 billion. Broadband speeds in wireless networks are expected to double from 1.7 Mbps to 4.0 Mbps to the end of 2019. As it is noted the mobile video traffic will be up to 72 percent of the global mobile traffic. As the number of wireless connections tends to increase significantly, it results in dramatic mobile traffic growth. Mobile service providers face the challenge to utilize limited radio resources efficiently. In this paper, we propose a mathematical model of radio resources allocation in broadband wireless networks such as LTE-Advanced in terms of queuing systems and evaluate blocking probability and average amount of occupied resources.

In this paper we concretize a condition when an aggregation of n oneserver queuing systems into multiserver system for $$n\rightarrow \infty $$n→∞ leads to an disappearance of a queue (in some probabilistic sense) and to a transformation of multiserver system into a system with infinite number of servers. An initial oneserver system is a system with Poisson input flow or with some modifications of this flow like a regular flow without an aftereffect or with Poisson flow in a random environment. Such formulation of a problem is connected with a large number of articles devoted to a modeling of computer networks by queuing systems with infinite number of servers and to a justification of these models application for real networks with finite number of servers.

Relevant to the modern theory of computer networks design questions of developing adequate service models of fractal traffic are considered in the article. The fidelity criteria of heavy-tailed distributions (HTD), which take into account the HTD distortion effect on the results of fractal queues simulation, are offered. The problem of HTD significant distortions in their realization during simulation is revealed and examined. To solve this problem we also developed the method, which does not require the use of “long arithmetic”.