The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

In a classic article, Granger (1966) asserted that most economic time series measured in level have spectra that exhibit a smooth declining shape with considerable power at very low frequencies. There has been no systematic attempt to examine Granger,s assertion with international data. We estimate output level spectra for 58 countries, divided into developed, high-income developing, and low-income developing groups. We find the shapes of the estimated spectra to be strikingly similar to Granger"s typical shape, particularly for the developed countries.

On the Typical Spectral Shape of an Economic Variable

This paper studies the customers’ equilibrium and socially optimal joining–balking behavior in single-server Markovian queues with multiple working vacations. Different from the classical vacation policies, the server does not completely stop service but maintains a low service rate in vacation state in case there are customer arrivals. Based on different precision levels of the system information, we discuss the observable queues, the partially observable queues, and the unobservable queues, respectively. For each type of queues, we get both the customers’ equilibrium and socially optimal joining–balking strategies and make numerical comparisons between them. We numerically observe that their equilibrium strategy is unique, and especially, the customers’ equilibrium joining probability in vacation state is not necessarily smaller than that in busy state in the partially observable queues. Moreover, we also find that the customers’ individual behavior always deviates from the social expectation and makes the system more congested.

Shiyong Li. Equilibrium and optimal behavior of customers in Markovian queues with multiple working vacations

Modelling the flexibility of production systems in Industry 4.0 for analysing their productivity and availability with high-level Petri nets

In this investigation, a single server M/M/1/N feedback queueing system with vacation, balking, reneging and retention of reneged customers is analyzed. By considering the mathematical modeling, we derive the steady state probabilities of the number of customers in the system. We obtain important measures of effectiveness of the model by using the stationary distribution, and develop a cost model of the queueing system. Further, a numerical study and a cost profit analysis are carried out.

Performance and economic analysis of a single server feedback queueing model with vacation and impatient customers

This paper presents an analysis of a Markovian feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption, where customers’ impatience is due to the servers’ vacation. The reneging times are assumed to be exponentially distributed. After the completion of service, each customer may reenter the system as a feedback customer for receiving another regular service with some probability or leave the system. A reneged customer can be retained in many cases by employing certain convincing mechanisms to stay in queue for completion of service. Thus, a reneged customer can be retained in the queueing system with some probability or he may leave the queue without receiving service. We establish the stationary analysis of the system. The probability generating functions of the stationary state probabilities is obtained, we deduce the explicit expressions of the system sizes when the server is in a normal service period and in a Bernoulli schedule vacation interruption, respectively. Various performance measures of the system are derived. Finally, we present some numerical examples to demonstrate how the various parameters of the model influence the behavior of the system.

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On feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption

This paper presents an analysis of heterogeneous two-server queueing system with feedback, reverse balking, reneging and retention of reneged customers. The steady-state solution of the model is obtained and different measures of effectiveness are derived. Economic analysis of the model is performed studying the impact of balking probability and retention probability on the total expected profit of the system.

Heterogeneous two-server queueing system with reverse balking and reneging

The present paper deals with an $$M^{X}/M/c$$
 Bernoulli feedback queueing system with variant multiple working vacations and impatience timers which depend on the states of the servers. Whenever a customer arrives at the system, he activates an random impatience timer. If his service has not been completed before his impatience timer expires, the customer may abandon the system. Using certain customer retention mechanism, the impatient customer can be retained in the system. After getting incomplete or unsatisfactory service, with some probability, each customer may comeback to the system as a Bernoulli feedback. Using the probability generating functions (PGFs), we derive the steady-state solution of the model. Then, we obtain useful performance measures. Moreover, we carry out an economic analysis. Finally, numerical study is performed to explore the effects of the model parameters on the behavior of the system.

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The $$M^{X}/M/c$$ M X / M / c Bernoulli feedback queue with variant multiple working vacations and impatient customers: performance and economic analysis

This paper deals with a finite capacity multi-server Markovian queueing model with Bernoulli feedback, synchronous multiple vacation policy and customers’ impatience (balking and reneging). By empl...

Amina Angelika Bouchentouf

Papers

Performance and economic analysis of a single server feedback queueing model with vacation and impatient customers

On feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption

Heterogeneous two-server queueing system with reverse balking and reneging

The $$M^{X}/M/c$$ M X / M / c Bernoulli feedback queue with variant multiple working vacations and impatient customers: performance and economic analysis

Analysis and Performance Evaluation of Markovian Feedback Multi-Server Queueing Model with Vacation and Impatience