Discrete-time renewal sequences play a fundamental role in the theory of stochastic processes. This article considers periodic versions of such processes; specifically, the length of an interrenewal is allowed to probabilistically depend on the season at which it began. Using only elementary renewal and Markov chain techniques, computational and limiting aspects of periodic renewal sequences are investigated. We use these results to construct a time series model for a periodically stationary sequence of integer counts.

We consider a general polling model with N stations. The stations are served exhaustively and in cyclic order. Once a station queue falls empty, the server does not immediately switch to the next station. Rather, it waits at the station for the possible arrival of new work (“wait-and-see”) and, in the case of this happening, it restarts service in an exhaustive fashion. The total time the server waits idly is set to be a fixed, deterministic parameter for each station. Switchover times and service times are allowed to follow some general distribution, respectively. In some cases, which can be characterized, this strategy yields a strictly lower average queuing delay than for the exhaustive strategy, which corresponds to setting the “wait-and-see credit” equal to zero for all stations. This extends the results of Peköz [12] and of Boxma et al. [4]. Furthermore, we give a lower bound for the delay for all strategies that allow the server to wait at the stations even though no work is present.

The total time on test transform (ttt) order was introduced in Kochar, Li, and Shaked (2002) for nonnegative random variables, which has a close connection to the location independent riskier and the excess wealth orders. In this article, the ttt order is redefined for comparing not necessarily nonnegative random variables. Such an extension will lead us to conveniently derive several new intrinsic properties of the ttt order. In particular, we establish an interesting separation result on the connection between the ttt and the excess wealth orders. As two applications of this separation result, we obtain the generating process of the ttt order in terms of mean-decreasing right stretches and the closure property of the ttt order under convolutions. The generating process of the ttt order parallels those of the location independent riskier and the excess wealth orders established by Landsberger and Meilijson (1994).

Here, we discuss the stochastic comparison of residual lifetimes of parallel systems and inactivity times of series systems by means of the reversed hazard rate order when the components of the systems are independent but not necessarily identically distributed. We also establish some monotonicity properties of such residual lifetimes of parallel systems and inactivity times of series systems. These results extend some of the recent results in this direction due to Zhao, Li, and Balakrishnan [21], Kochar and Xu [12], and Saledi and Asadi [16].

When launching a new product, a firm has to set an initial price with incomplete information about demand. However, after observing the demand over a period of time, the firm might decide to mark down the price, especially when the Bayesian updated belief about demand is lower than originally anticipated. We consider the case in which the manufacturer makes three decisions: initial price, when to mark down the price, and the markdown price. Modeling the cumulative demand as a Brownian motion with an unknown drift, we compute the posterior probability distribution of the unknown drift. We then show that it is optimal to mark down the price when the posterior probability is below a computable threshold. This threshold policy enables us to determine the optimal (a) regular price, (b) markdown price, and (c) markdown time. Additionally, we examine the impact of demand volatility and evaluate the value of learning.

The speed of the random walk on a tree is the rate of escaping its starting point. It depends on the way that the branching occurs in the sense that if the average number of branching is large, the speed is more likely to be positive. The speed on some models of random trees is calculated via calculating the hitting times of the consecutive levels of the tree.

We consider binary coherent systems with independent binary components having equal failure probability q. The system DOWN probability is expressed via its signature's combinatorial analogue, the so-called D-spectrum. Using the definition of the Birnbaum importance measure (BIM), we introduce for each component a new combinatorial parameter, so-called BIM-spectrum, and develop a simple formula expressing component BIM via the component BIM-spectrum. Further extension of this approach allows obtaining a combinatorial representation for the joint reliability importance (JRI) of two components. To estimate component BIMs and JRIs, there is no need to know the analytic formula for system reliability. We demonstrate how our method works using the Monte Carlo approach. We present several examples of estimating component importance measures in a network when the DOWN state is defined as the loss of terminal connectivity.

In this article we present a sequence of random variables with weighted tail distribution functions, constructed based on the relevation transform. For this sequence, we prove several recursive formulas and connections to the residual entropy through the unifying framework of the Dickson–Hipp operator. We also give some numerical examples to evaluate our results.