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This paper models a cash conversion system in a subsidiary of a parent company where there is an active internal capital market, but otherwise the subsidiary has no access to additional external funds. The cash conversion system consists of a treasury, a single-product make-to-stock inventory, and a receivables pool. It implements a perpetual flow cycle, where funds convert to product and back to funds. The system is stationary, and revenues and costs flow directly to the parent company. The parent company aims to maximize equilibrium (long-run) financial metrics in terms of net profit rate and rate of return. To this end, we model this system as a discrete-state continuous-time Markov process and compute its equilibrium state distribution using analytic and numerical methods. These are then used to derive statistics of the equilibrium cash conversion cycle and define equilibrium financial rate metrics and cumulative counterparts that incorporate the time value of money. We further optimize the financial and operational designs of the system and, specifically, the internal capital allocation and inventory base stock level. Finally, noting the potential for friction in the parent–subsidiary relationship, we study numerically the impact of moral hazard and internal capital market inefficiency on optimal designs. The online appendix is available at https://doi.org/10.1287/msom.2017.0625 .

This paper considers a continuous-review, single-product, production-inventory system with a constant replenishment rate, compound Poisson demands, and lost sales. Two objective functions that represent metrics of operational costs are considered: (1) the sum of the expected discounted inventory holding costs and lost-sales penalties, both over an infinite time horizon, given an initial inventory level; and (2) the long-run time average of the same costs. The goal is to minimize these cost metrics with respect to the replenishment rate. It is, however, not possible to obtain closed-form expressions for the aforementioned cost functions directly in terms of positive replenishment rate (PRR). To overcome this difficulty, we construct a bijection from the PRR space to the space of positive roots of Lundberg's fundamental equation, to be referred to as the Lundberg positive root (LPR) space. This transformation allows us to derive closed-form expressions for the aforementioned cost metrics with respect to the LPR variable, in lieu of the PRR variable. We then proceed to solve the optimization problem in the LPR space and, finally, recover the optimal replenishment rate from the optimal LPR variable via the inverse bijection. For the special cases of constant or loss-proportional penalty and exponentially distributed demand sizes, we obtain simpler explicit formulas for the optimal replenishment rate.

This paper studies the spectral properties of a class of stochastic sequences generated by basic TES (Transform-Expand-Sample) methods, as well as transformed TES sequences. A TES sequence is obtained as a marginally uniform process from an autoregressive scheme with modulo-1 reduction, followed by additional transformations. We derive formulas for the spectral density function and the spectral distribution function which are suitable for efficient numerical computation. A set of examples is calculated and exhibited for TES processes with uniform and exponential marginals. The results contribute to the understanding of TES sequences as models of autocorrelated sequences, particularly in a Monte Carlo simulation context. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

In this paper we establish a joint central limit theorem for customer and time averages by applying a martingale central limit theorem in a Markov framework. The limiting values of the two averages appear in the translation terms. This central limit theorem helps to construct confidence intervals for estimators and perform statistical tests. It thus helps determine which finite average is a more asymptotically efficient estimator of its limit. As a basis for testing for PASTA (Poisson arrivals see time averages), we determine the variance constant associated with the central limit theorem for the difference between the two averages when PASTA holds.

This paper introduces a class of methods called TES (Transform-Expand-Sample) for generating autocorrelated variates with uniform marginals and Markovian structure. TES methods are readily implemented on a computer and have generation complexity comparable to that of the i.i.d. uniform sequence which they transform to an autocorrelated uniform sequence. For any prescribed correlation coefficient ρ, there is a TES method generating a uniform sequence with the 1-lag autocorrelation ρ, and the resultant autocorrelation is monotonic quadratic in two structural TES parameters. A simulation study reveals that TES methods give rise to autocorrelation functions with monotone decreasing as well as oscillating magnitude, bounded by monotone envelopes. The structural parameters were found to control the “amplitude” and “frequency” of the resultant autocorrelation function. A third parameter can be used to transform a TES sequence into more continuous-looking versions and to control the skewness of sample path cycles. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

We investigate when Arrivals See Time Averages (ASTA) in a stochastic model; i.e., when the stationary distribution of an embedded sequence, obtained by observing a continuous-time stochastic process just prior to the points (arrivals) of an associated point process, coincides with the stationary distribution of the observed process. We also characterize the relation between the two distributions when ASTA does not hold. We introduce a Lack of Bias Assumption (LBA) which stipulates that, at any time, the conditional intensity of the point process, given the present state of the observed process, be independent of the state of the observed process. We show that LBA, without the Poisson assumption, is necessary and sufficient for ASTA in a stationary process framework. Consequently, LBA covers known examples of non-Poisson ASTA, such as certain flows in open Jackson queueing networks, as well as the familiar Poisson case (PASTA). We also establish results to cover the case in which the process is observed just after the points, e.g., when departures see time averages. Finally, we obtain a new proof of the Arrival Theorem for product-form queueing networks.

PASTA (Poisson Arrivals See Time Averages) is a term coined by R. Wolff in his well known 1982 paper. In keeping with Wolff's terminology, we use the term anti-PASTA to refer to the following converse of PASTA. Given that arrivals do indeed see time averages, when must the arrival process necessarily be Poisson? We show that anti-PASTA is satisfied in a pure-jump Markov process, provided that the arrival process corresponds to a subset of the Markov process jumps.

Burke and Reich independently showed that the output of an M/M/1 queue in equilibrium is a Poisson process. Consequently, analysis of series (tandem) exponential servers with a Poisson input stream can be reduced to consideration of a series of M/M/1 queues. This work generalizes the above results to so-called Jackson networks, consisting of exponential servers with mutually independent Poisson exogenous inputs and random customer routings permitting customer feedback. We prove that traffic on all exit arcs of a network in equilibrium is Poisson; moreover, the customer streams leaving any exit set are mutually independent. Here an exit arc is a path from server node i such that a customer moving along the arc cannot return to i; an exit set V is a set of server nodes such that customers departing V can never re-enter V. As a special case, the traffic streams leaving a Jackson network in equilibrium are mutually independent Poisson processes. In contrast, traffic on non-exit arcs is non-Poisson, and indeed non-renewal. A canonical decomposition of the server nodes is defined as a partition of the server node set into components C*q, each consisting of communicating servers. We show that the set of all exit arcs coincides with the arcs emanating from the C*q. Finally, for a Jackson network in equilibrium, each C*q is itself a Jackson network in equilibrium.

We consider a queueing network with types of customers. Poisson arrivals, type dependent routing and state dependent service. A number of stochastic aspects relating to sojourn times are investigated. Included are limiting state distributions imbedded at customer move times, mean sojourn times, and sojourn time distributions along overtake-free paths. It is shown that Reich's results extend to overtake-free paths yielding limiting sojourn time distributions which correspond to the sum of exponential independent stages of sojourn in individual nodes. The analysis is bused solely on considerations of population distributions through conditioning on customers' itineraries in the network.

This paper proposes a methodology for computing numerical bounds on cumulative-time (i.e., the time spent in a specified set of states before entering another specified set of states) distributions over discrete state Markov processes. The methodology uses randomization procedures to compute results for appropriately defined transient Markov processes; the transient distribution computed is equivalent to the requisite cumulative-time distribution. A queueing application of the methodology to delay times in queueing networks is outlined and its efficacy is appraised by comparing the results with a sojourn time for a problem with a known distribution.