M/M/1 Queue with Negative Customers

2019 ◽  
Vol 08 (01) ◽  
pp. 165-175
Author(s):  
金萍 徐
Keyword(s):  
Negative Customers

State-dependent signalling in queueing networks

1994 ◽  
Vol 26 (02) ◽  
pp. 436-455 ◽  
Author(s):  
W. Henderson ◽  
B. S. Northcote ◽  
P. G. Taylor
Keyword(s):  
Queueing Networks ◽  
State Dependence ◽  
Product Form ◽  
Batch Size ◽  
Single Server ◽  
Negative Customers ◽  
Affect Control ◽  
State Dependent ◽  
Special Cases ◽  
Equilibrium Distributions

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network. This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.

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

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Doo Ho Lee
Keyword(s):  
Queueing System ◽  
Point Of View ◽  
Optimal Pricing ◽  
Pricing Strategies ◽  
Negative Customers ◽  
Economic Point ◽  
Equilibrium Behavior ◽  
Payment Scheme ◽  
Ex Post ◽  
Pricing Schemes

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 QUEUE WITH POSITIVE AND NEGATIVE ARRIVALS GOVERNED BY A MARKOV CHAIN

2003 ◽  
Vol 17 (4) ◽  
pp. 487-501 ◽  
Author(s):  
Yang Woo Shin ◽  
Bong Dae Choi
Keyword(s):  
Markov Chain ◽  
Transient State ◽  
Single Server ◽  
Negative Customers ◽  
Service Distribution ◽  
Finite State ◽  
Sojourn Time Distribution ◽  
Absorbing States ◽  
Negative Arrivals ◽  
Selection Of

We consider a single-server queue with exponential service time and two types of arrivals: positive and negative. Positive customers are regular ones who form a queue and a negative arrival has the effect of removing a positive customer in the system. In many applications, it might be more appropriate to assume the dependence between positive arrival and negative arrival. In order to reflect the dependence, we assume that the positive arrivals and negative arrivals are governed by a finite-state Markov chain with two absorbing states, say 0 and 0′. The epoch of absorption to the states 0 and 0′ corresponds to an arrival of positive and negative customers, respectively. The Markov chain is then instantly restarted in a transient state, where the selection of the new state is allowed to depend on the state from which absorption occurred.The Laplace–Stieltjes transforms (LSTs) of the sojourn time distribution of a customer, jointly with the probability that the customer completes his service without being removed, are derived under the combinations of service disciplines FCFS and LCFS and the removal strategies RCE and RCH. The service distribution of phase type is also considered.

The M/G/1 queue with negative customers

1996 ◽  
Vol 28 (02) ◽  
pp. 540-566 ◽  
Author(s):  
Peter G. Harrison ◽  
Edwige Pitel
Keyword(s):  
Iterative Solution ◽  
Single Server ◽  
Negative Customers ◽  
Poisson Arrival ◽  
First Case ◽  
Full Study ◽  
Service Times ◽  
The Stability ◽  
The One ◽  
Iterative Solution Method

We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer—the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

scholarly journals Perishable Inventory System with Postponed Demands and Negative Customers

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
Paul Manuel ◽  
B. Sivakumar ◽  
G. Arivarignan
Keyword(s):  
Steady State ◽  
Joint Probability ◽  
Inventory System ◽  
Arrival Process ◽  
Inventory Level ◽  
Joint Probability Distribution ◽  
Time Interval ◽  
Negative Customers ◽  
Perishable Inventory System ◽  
Number Of Customers

This article considers a continuous review perishable (s,S) inventory system in which the demands arrive according to a Markovian arrival process (MAP). The lifetime of items in the stock and the lead time of reorder are assumed to be independently distributed as exponential. Demands that occur during the stock-out periods either enter a pool which has capacity N(<∞) or are lost. Any demand that takes place when the pool is full and the inventory level is zero is assumed to be lost. The demands in the pool are selected one by one, if the replenished stock is above s, with time interval between any two successive selections distributed as exponential with parameter depending on the number of customers in the pool. The waiting demands in the pool independently may renege the system after an exponentially distributed amount of time. In addition to the regular demands, a second flow of negative demands following MAP is also considered which will remove one of the demands waiting in the pool. The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady state case. The measures of system performance in the steady state are calculated and the total expected cost per unit time is also considered. The results are illustrated numerically.

EQUILIBRIUM BALKING STRATEGIES IN THE REPAIRABLE M/M/1 G-RETRIAL QUEUE WITH COMPLETE REMOVALS

2019 ◽  
pp. 1-20
Author(s):  
Shan Gao ◽  
Deran Zhang ◽  
Hua Dong ◽  
Xianchao Wang
Keyword(s):  
Nash Equilibrium ◽  
Queueing Theory ◽  
Fundamental Problem ◽  
Pure Strategy ◽  
Retrial Queue ◽  
Noncooperative Game ◽  
Net Benefit ◽  
Negative Customers ◽  
Stable Strategy ◽  
Best Response

We consider an M/M/1 retrial queue subject to negative customers (called as G-retrial queue). The arrival of a negative customer forces all positive customers to leave the system and causes the server to fail. At a failure instant, the server is sent to be repaired immediately. Based on a natural reward-cost structure, all arriving positive customers decide whether to join the orbit or balk when they find the server is busy. All positive customers are selfish and want to maximize their own net benefit. Therefore, this system can be modeled as a symmetric noncooperative game among positive customers and the fundamental problem is to identify the Nash equilibrium balking strategy, which is a stable strategy in the sense that if all positive customers agree to follow it no one can benefit by deviating from it, that is, it is a strategy that is the best response against itself. In this paper, by using queueing theory and game theory, the Nash equilibrium mixed strategy in unobservable case and the Nash equilibrium pure strategy in observable case are considered. We also present some numerical examples to demonstrate the effect of the information together with some parameters on the equilibrium behaviors.

Negative Customers Model Queues with Breakdowns

1996 ◽  
pp. 153-167
Author(s):  
Peter G. Harrison ◽  
Edwige Pitel ◽  
Naresh M. Patel
Keyword(s):  
Negative Customers

ANALYSIS OF THE NETWORK WITH MULTIPLE CLASSES OF POSITIVE AND NEGATIVE CUSTOMERS AT A TRANSIENT REGIME

2018 ◽  
Vol 33 (2) ◽  
pp. 172-185 ◽  
Author(s):  
Mikhail Matalytski
Keyword(s):  
Transient Regime ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Stationary Probability Distribution ◽  
Negative Customers ◽  
Network Systems ◽  
Modified Method ◽  
Computer Calculations ◽  
Network States ◽  
The Impact

This paper is devoted to the investigation of the G-network with multiple classes of positive and negative customers. The purpose of the investigation is to analyze such a network at a transient regime, to find the state probabilities of the network that depend on time. In the first part, a description of the functioning of G-networks with positive and negative customers is provided, when a negative customer when arriving to the system destroys a positive customer of its class. Streams of positive and negative customers arriving at 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 nonstationary probabilities of network states, a system of Kolmogorov's difference-differential equations (DDE) 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. It is proved that successive approximations converge with time to a stationary probability distribution, the form of which is indicated in this paper, and the sequence of approximations converges to the unique solution of the DDE system. Any successive approximation is representable in the form of a convergent power series with an infinite radius of convergence, the coefficients of which satisfy recurrence relations, which is convenient for computer calculations. A model example illustrating the determination of the time-dependent probabilities of network states using this technique has been calculated. The obtained results can be applied in modeling the behavior of computer viruses and attacks in information and telecommunication systems and networks, for example, as a model of the impact of some file viruses on server resources. variable.

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