Negative Customers Research Articles

FINDING EXPECTED REVENUES IN G-NETWORK WITH MULTIPLE CLASSES OF POSITIVE AND NEGATIVE CUSTOMERS

Investigation of the G-network with multiple classes of positive and negative customers has been carried out in the article. The purpose of the investigation is to analyze such a network at a transient regime, finding expected revenues in the network systems depending on time. A negative customer arriving to the system and destroys a positive customer of its class. Streams of positive and negative customers arriving to each of the network systems are independent. Services of positive customers of all types occur in accordance with a random selection of them for service. For the expected revenues, a system of Kolmogorov's difference-differential equations has been derived. A method for their finding is proposed. It is based on the use of a modified method of successive approximations, combined with the method of series. A model example illustrating the finding of time-dependent expected revenues of network systems has been calculated, which shows that the expected revenues of network systems can be either increasing or decreasing time functions. The obtained results can be applied in forecasting losses in information and telecommunication systems and networks from the penetration of computer viruses into it and conducting computer attacks.

Анализ в переходном режиме сети с нетерпеливыми положительными и отрицательными заявками различных типов

Анализ в переходном режиме сети с нетерпеливыми положительными и отрицательными заявками различных типов

A study on M/G/1 retrial queueing system with three different types of customers under working vacation policy

This paper deals with a single server retrial queueing system with working vacations and vacation interruption. There are three different types of customers are considered, which are priority customers, ordinary customers and negative customers. The priority customers do not form any queue and have an exclusive preemptive priority to receive their services over ordinary customers. The negative customer is arriving during the service time of any positive customer, will remove the positive customer from the service. If the interrupted customer is an ordinary customer, he will leave the system. As soon as the orbit becomes empty at the time of service completion, the server goes for a working vacation. The server works at a lower speed during a working vacation period. Using the supplementary variable technique, the steady-state probability generating a function of the system and its orbit are found. Some numerical examples are presented.

A study on M/G/1 retrial queueing system with three different types of customers under working vacation policy

This paper deals with a single server retrial queueing system with working vacations and vacation interruption. There are three different types of customers are considered, which are priority customers, ordinary customers and negative customers. The priority customers do not form any queue and have an exclusive preemptive priority to receive their services over ordinary customers. The negative customer is arriving during the service time of any positive customer, will remove the positive customer from the service. If the interrupted customer is an ordinary customer, he will leave the system. As soon as the orbit becomes empty at the time of service completion, the server goes for a working vacation. The server works at a lower speed during a working vacation period. Using the supplementary variable technique, the steady-state probability generating a function of the system and its orbit are found. Some numerical examples are presented.

Stationary queue length of a single-server queue with negative arrivals and nonexponential service time distributions

In this paper, a single-server queue with negative customers is considered. The arrival of a negative customer will remove one positive customer that is being served, if any is present. An alternative approach will be introduced to derive a set of equations which will be solved to obtain the stationary queue length distribution. We assume that the service time distribution tends to a constant asymptotic rate when time t goes to infinity. This assumption will allow for finding the stationary queue length of queueing systems with non-exponential service time distributions. Numerical examples for gamma distributed service time with fractional value of shape parameter will be presented in which the steady-state distribution of queue length with such service time distributions may not be easily computed by most of the existing analytical methods.

On the multiserver retrial queues with negative arrivals.

The paper deals with an M/M/C/K retrial queue with exponential abandonment at which positive and negative customers arrive according to Poisson processes. This model is of practical interest: it can be used for analysing the performance in call centres and e-mail contact systems. For the model under investigation, we find the ergodicity conditions and also the approximate solution by using finite truncation and value extrapolation methods. We present some numerical results to examine the performance of the value extrapolation method as well as the system performance.

A group arrival retrial G - queue with multi optional stages of service, orbital search and server breakdown

A group arrival feedback retrial queue with k optional stages of service and orbital search policy is studied. Any arriving group of customer finds the server free, one from the group enters into the first stage of service and the rest of the group join into the orbit. After completion of the ith stage of service, the customer under service may have the option to choose (i+1)th stage of service with θi probability, with pI probability may join into orbit as feedback customer or may leave the system with probability. Busy server may get to breakdown due to the arrival of negative customers and the service channel will fail for a short interval of time. At the completion of service or repair, the server searches for the customer in the orbit (if any) with probability α or remains idle with probability 1-α. By using the supplementary variable method, steady state probability generating function for system size, some system performance measures are discussed.

A study on M/G/1 retrial G - queue with two phases of service, immediate feedback and working vacations

In this paper, we discuss about the steady state behaviour of M/G/1 retrial queueing system with two phases of services and immediate feedbacks under working vacation policy where the regular busy server is affected due to the arrival of negative customers. Upon arrival if the customer finds the server busy, breakdown or on working vacation it enters an orbit; otherwise the customer enters into the service area immediately. After service completion, the customer is allowed to make finite number of immediate feedback. The feedback service also consists of two phases. At the service completion epoch of a positive customer, if the orbit is empty the server goes for a working vacation. The server works at a lower service rate during working vacation (WV) period. Using the supplementary variable technique, we found out the steady state probability generating function for the system and in orbit. System performance measures and reliability measures are discussed. Finally, some numerical examples are presented to validate the analyticalresults.

Stationary probability distribution for states of G-networks with constrained sojourn time

We consider an exponential queueing network that differs from a Gelenbe network (with the usual positive and so-called negative customers), first, in that the sojourn time of customers at the network nodes is bounded by a random value whose conditional distribution for a fixed number of customers in a node is exponential. Second, we significantly relax the conditions on possible values of parameters for incoming Poisson flows of positive and negative customers in Gelenbe’s theorem. Claims serviced at the nodes and customers leaving the nodes at the end of their sojourn time can stay positive, become negative, or leave the network according to different routing matrices. We prove a theorem that generalizes Gelenbe’s theorem.

G-Networks with Adders

Abstract: Queueing networks are used to model the performance of the Internet, of manufacturing and job-shop systems, supply chains, and other networked systems in transportation or emergency management. Composed of service stations where customers receive service, and then move to another service station till they leave the network, queueing networks are based on probabilistic assumptions concerning service times and customer movement that represent the variability of system workloads. Subject to restrictive assumptions regarding external arrivals, Markovian movement of customers, and service time distributions, such networks can be solved efficiently with “product form solutions” that reduce the need for software simulators requiring lengthy computations. G-networks generalise these models to include the effect of “signals” that re-route customer traffic, or negative customers that reject service requests, and also have a convenient product form solution. This paper extends G-networks by including a new type of signal, that we call an “Adder”, which probabilistically changes the queue length at the service center that it visits, acting as a load regulator. We show that this generalisation of G-networks has a product form solution.

Forecasting anticipated incomes in the Markov networks with positive and negative customers

A method for determination of the time dependence of the anticipated income in the systems of Markov queuing networks with incomes and positive and negative customers was proposed. Subject to this proviso, the incomes from the transitions between the network states are deterministic functions depending on its state and time, and the system incomes in a time unit when they do not change their states depend only on these states. An illustrative example of calculations was given which shows that the anticipated incomes of the network systems can be both increasing and decreasing time functions assuming positive and negative values.

ANALYSIS OF MULTI-RESOURCE LOSS SYSTEM WITH STATE-DEPENDENT ARRIVAL AND SERVICE RATES

In this paper, we study a generalization of the classical multi-dimensional Erlang loss model with state-dependent arrival and service rates, in which customers at arrival occupy random quantities of various resources and release them at departure. Total amount of resources allocated to customers cannot exceed predefined maximum levels. There can be two types of customers: positive customers, which occupy positive quantities of resources, and negative customers, which occupy negative quantities of resources. Negative customers increase the amount of resources available to positive customers and therefore decrease blocking of positive customers caused by lack of resources.

Analysis of the queueing network with a random bounded waiting time of positive and negative customers at a non-stationary regime

Matalytski, Mikhail Analysis of the queueing network with a random bounded waiting time of positive and negative customers at a non-stationary regime/M. Matalytski, V. Naumenko // Jornal of Applied Mathematics and Computational Mechanics.- 2017 .- Т.16. - № 1.- С.97-108 Статья

An interdiction game on a queueing network with multiple intruders

Security forces are deployed to protect networks that are threatened by multiple intruders. To select the best deployment strategy, we analyze an interdiction game that considers multiple simultaneous threats. Intruders route through the network as regular customers, while interdictors arrive at specific nodes as negative customers. When an interdictor arrives at a node where an intruder is present, the intruder is removed from the network. Intruders and interdictors compete over the value of this network, which is the throughput of unintercepted intruders. Intruders attempt to maximize this throughput by selecting a fixed route through the network, while the interdictors aim to minimize the throughput selecting their arrival rate at each node. We analyze this game and characterize optimal strategies. For special cases, we obtain explicit formulas to evaluate the optimal strategies and use these to compute optimal strategies for general networks. We also consider the network with probabilistic routing of intruders and show that for this case, the value and optimal strategies of the interdictor of the resulting game remain the same.

Retrial inventory model with negative customers and multiple working vacations

This paper analyses a continuous review (s, S) inventory system at a service facility, wherein an item demanded by a customer is issued after performing service on the item. Primary arrivals follow a Poisson distribution and they may turn out to be positive or negative and then enter into the orbit. The orbiting demands compete for service according to an exponential distribution. The server takes multiple working vacations at zero inventory. Replenishment times, vacation times, service times during regular busy periods and vacation periods are exponentially distributed. A matrix analytic method is used for the steady-state distribution of the model. Various performance measures and cost analyses are shown with numerical results.

Whittle-networks with signals

We present an extension of Whittle networks with customers and signals as defined by Gelenbe in his seminal paper on negative customers. Customers are queued and served according to the balance rules defined for Whittle networks while signals are not queued but interact with the customers at their arrival. We consider various types of signals: they can delete customers, flush out a queue or change the class of a customer. We prove that all these networks have a product form steady-state distribution as soon as the signals obey a balance rule consistent with the service balance defined by Whittle.

A new type of negative customers

The possibility of removing a customer before being served is one that is often absent in the theory of waiting queues. This reminds us of the concept of negative customer, which was set forth by three leading scientists: Gelenbe, Boucherie-van Dijk and Henderson. Negative customers are likely to eliminate positive ones and make them disappear. They cannot build up in the queue, and are simply lost if they arrive to an empty queue (Gelenbe, 1991; Gelenbe et al., 1991). In other contributions (Boucherie and van Dijk, 1994), we find a type of negative customers that can build up in the queue, as opposed to Gelenbe negative customers. We introduce a new type of negative client, having the same properties mentioned above, but characterised by the fact that they may remove one or more regular clients, and leave the system after being server as a regular customer. We study the existence of stationary distribution, the intensity of transition, along with the quasi-reversibility of this new type of negative customer.

Analysis of a preemptive priority retrial queue with negative customers, starting failure and at most J vacations

An analysis of single server preemptive priority retrial queue with at most J vacations where two types of customers called (priority customers and ordinary customers) are considered in this paper. The priority customers do not have queue and they have higher priority to receive their services over ordinary customers. If negative customer is arriving during the service time of any positive customer (priority customer or ordinary customer), it will remove the positive customer from the service. If the interrupted customer is an ordinary customer, he may join the orbit and the priority customer will leave the system. As soon as the system is empty, the server takes at most J vacations. The probability generating functions for the system/orbit size in steady state is obtained using supplementary variable method. Some important system measures and the stochastic decomposition are discussed. Numerical examples are presented to picturise the effect of parameters on system performance measures.

Optimal Pricing Strategies and Customers’ Equilibrium Behavior in an Unobservable M/M/1 Queueing System with Negative Customers and Repair

This work investigates the optimal pricing strategies of a server and the equilibrium behavior of customers in an unobservable M/M/1 queueing system with negative customers and repair. In this work, we consider two pricing schemes. The first is termed the ex‐post payment scheme, where the server charges a price that is proportional to the time spent by a customer in the system. The second scheme is the ex‐ante payment scheme, where the server charges a flat rate for all services. Based on the reward‐cost structure, the server (or system manager) should make optimal pricing decisions in order to maximize its expected profit per time unit in each payment scheme. This study also investigates equilibrium joining/balking behavior under the server’s optimal pricing strategies in the two pricing schemes. We show, given a customer’s equilibrium, that the two pricing schemes are perfectly identical from an economic point of view. Finally, we illustrate the effect of several system parameters on the optimal joining probabilities, the optimal price, and the equilibrium behavior via numerical examples.

A new type of negative customers

The possibility of removing a customer before being served is one that is often absent in the theory of waiting queues. This reminds us of the concept of negative customer, which was set forth by three leading scientists: Gelenbe, Boucherie-van Dijk and Henderson. Negative customers are likely to eliminate positive ones and make them disappear. They cannot build up in the queue, and are simply lost if they arrive to an empty queue (Gelenbe, 1991; Gelenbe et al., 1991). In other contributions (Boucherie and van Dijk, 1994), we find a type of negative customers that can build up in the queue, as opposed to Gelenbe negative customers. We introduce a new type of negative client, having the same properties mentioned above, but characterised by the fact that they may remove one or more regular clients, and leave the system after being server as a regular customer. We study the existence of stationary distribution, the intensity of transition, along with the quasi-reversibility of this new type of negative customer.