Search

Significance Donald Trump's 2016 victory in the Electoral College without leading in the popular vote has led to speculation of a repeat in 2020. If the Democrats were to win the popular vote but by less than a landslide, could they be thwarted from stopping Trump once again? We examine how Electoral College outcomes are conditioned by how states had voted in previous elections. Our simulations suggest that in 2020, the Electoral College bias is likely to again favor the Republicans. However, the expectation is less distortion of the popular will than in 2016. There is even the possibility that if Trump was to win the 2020 popular vote by a slim margin, it could be he who loses the Electoral College., Donald Trump’s 2016 win despite failing to carry the popular vote has raised concern that 2020 would also see a mismatch between the winner of the popular vote and the winner of the Electoral College. This paper shows how to forecast the electoral vote in 2020 taking into account the unknown popular vote and the configuration of state voting in 2016. We note that 2016 was a statistical outlier. The potential Electoral College bias was slimmer in the past and not always favoring the Republican candidate. We show that in past presidential elections, difference among states in their presidential voting is solely a function of the states’ most recent presidential voting (plus new shocks); earlier history does not matter. Based on thousands of simulations, our research suggests that the bias in 2020 probably will favor Trump again but to a lesser degree than in 2016. The range of possible outcomes is sufficiently wide, however, to even include some possibility that Joseph Biden could win in the Electoral College while barely losing the popular vote.

Using a result of Blanchet and Wallwater (2015: Exact sampling of stationary and time-reversed queues. ACM TOMACS, 25, 26) for exactly simulating the maximum of a negative drift random walk queue endowed with independent and identically distributed (iid) increments, we extend it to a multi-dimensional setting and then we give a new algorithm for simulating exactly the stationary distribution of a first-in-first-out (FIFO) multi-server queue in which the arrival process is a general renewal process and the service times are iid; the FIFO GI/GI/c queue with 2 \le c < 1. Our method utilizes dominated coupling from the past (DCFP) as well as the Random Assignment (RA) discipline, and complements the earlier work in which Poisson arrivals were assumed, such as the recent work of Connor and Kendall (2015: Perfect simulation of M/G/c queues. Advances in Applied Probability, 47, 4). We also consider the models in continuous-time and show that with mild further assumptions, the exact simulation of those stationary distributions can also be achieved. We also give, using our FIFO algorithm, a new exact simulation algorithm for the stationary distribution of the infinite server case, the GI/GI/\infty model. Finally, we even show how to handle Fork-Join queues, in which each arriving customer brings c jobs, one for each server., 32 pages, 4 figures

We develop a general framework for stationary marked point processes in discrete time. We start with a careful analysis of the sample paths. Our initial representation is a sequence $$\{(t_j,k_j): j\in {\mathbb {Z}}\}$$ of times $$t_j\in {\mathbb {Z}}$$ and marks $$k_j\in {\mathbb {K}}$$ , with batch arrivals (i.e., $$t_j=t_{j+1}$$ ) allowed. We also define alternative interarrival time and sequence representations and show that the three different representations are topologically equivalent. Then, we develop discrete analogs of the familiar stationary stochastic constructs in continuous time: time-stationary and point-stationary random marked point processes, Palm distributions, inversion formulas and Campbell’s theorem with an application to the derivation of a periodic-stationary Little’s law. Along the way, we provide examples to illustrate interesting features of the discrete-time theory.

We consider a single-item continuous-review (r, q) inventory system with a renewal demand process and independent, identically distributed stochastic lead times. Using a stationary marked-point process technique and a heavy-traffic limit, we prove a previous conjecture that inventory position and inventory on-order are asymptotically independent. We also establish closed-form expressions for the optimal policy parameters and system cost in heavy-traffic limit, the first of their kind, to our knowledge. These expressions sharpen our understanding of the key determinants of the optimal policy and their quantitative and qualitative impacts. For example, the results demonstrate that the well-known square-root relationship between the optimal order quantity and demand rate under a sequential processing environment is replaced by the cube root under a stochastic parallel processing environment. We further extend the study to periodic-review (S, T) systems with constant lead times. The electronic companion is available at https://doi.org/10.1287/opre.2017.1623 .

In Sigman (J. Appl. Probab. 48A:209–216, 2011b), a first exact simulation algorithm was presented for the stationary distribution of customer delay for FIFO M/G/c queues in which ρ=λ/μ

We present an exact simulation algorithm for the stationary distribution of customer delay for FIFO M/G/c queues in which ?=?/μ y)\,dy$ are such that samples of them can be simulated. Unlike the methods used in Sigman (J. Appl. Probab. 48A:209---216, 2011) involving coupling from the past, here we use different methods involving discrete-time processes and basic regenerative simulation, in which, as regeneration points, we use return visits to state 0 of a corresponding random assignment (RA) model which serves as a sample-path upper bound.

We establish heavy-traffic limits for stationary waiting times and other performance measures in G n /G n /1 queues, where G n indicates that an original point process is modified by cyclic thinning of order n, i.e., the thinned process contains every nth point from the original point process. The classical example is the Erlang E n /E n /1 queue, where cyclic thinning of order n is applied to both the interarrival times and the service times, starting from a "base" M/M/1 model. The models G n /D/1 and D/G n /1 are special cases of G n /G n /1. Since waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated G n /D/1 model, where the interarrival times are the sum of n consecutive interarrival times in the original model, the G/D/n model is a special case as well. As n??, the G n /G n /1 models approach the deterministic D/D/1 model. We obtain revealing limits by letting ? n ?1 as n??, where ? n is the traffic intensity in model n.

We present an analysis of labor mobility as a predictor of wages and job turnover. Data from the National Longitudinal Surveys of Youth show that workers with a history of less frequent job changes (stayers) earn higher wages and change jobs less frequently in the future than their more mobile counterparts (movers). These mobility effects on wages and turnover are stronger among more experienced workers, are highly robust across various model specifications, and persist despite corrections for unobserved individual fixed effects. In the second half of the paper we present a simple two period stochastic model of job mobility to study wages across movers and stayers. The model, incorporating salient features of human capital and job search, shows that whether stayers earn more than movers depend on the distribution of outside wage offers and firm-specific wage growth rate. Incorporating heterogeneity of wage growth rates among jobs increases the likelihood that stayers earn more than movers. D 2003 Published by Elsevier B.V.

Using a known fact that a Galton–Watson branching process can be represented as an embedded random walk, together with a result of Heyde (1964), we first derive finite exponential moment results for the total number of descendants of an individual. We use this basic and simple result to prove analogous results for the population size at time t and the total number of descendants by time t in an age-dependent branching process. This has applications in justifying the interchange of expectation and derivative operators in simulation-based derivative estimation for generalized semi-Markov processes. Next, using the result of Heyde (1964), we show that, in a stable GI/GI/1 queue, the length of a busy period and the number of customers served in a busy period have finite exponential moments if and only if the service time does.

We explore the performance of an M/GI/1 queue under various scheduling policies from the perspective of a new metric: the slowdown experienced by the largest jobs. We consider scheduling policies that bias against large jobs, towards large jobs, and those that are fair, e.g., processor-sharing (PS). We prove that as job size increases to infinity, all work conserving policies converge almost surely with respect to this metric to no more than 1/(1 - ρ), where ρ denotes the load. We also find that the expected slowdown under any work conserving policy can be made arbitrarily close to that under PS, for all job sizes that are sufficiently large.

The mathematical theory of congestion is associated with delays while waiting in a line or queue for service in a system. Examples of such systems include banks, post offices, and supermarkets as well as telecommunication systems involving telephones, computer networks, the Internet/the World Wide Web, inventory, and manufacturing processes. The early development of the field was motivated by telephone systems and the ways in which they could be made to operate more efficiently. Much of the theory relies heavily on the use of probability theory and stochastic processes, and as such is widely viewed as a subfield of stochastic modeling and applied probability. There is also significant interplay with other fields such as scheduling theory, inventory theory, and insurance risk theory; it is a central part of operations research. Keywords: queueing-risk duality; mathematical congestion; queueing networks; queueing theory; ruin probabilities; stochastic recursions; sojourn times

We consider a single-item continuous-review (r,q) inventory system with a renewal demand process and i.i.d. stochastic leadtimes. Using a stationary marked point process technique and a heavy traffic limit, we prove a previous conjecture that inventory position and inventory on-order are asymptotically independent. We also establish closed-form expressions for the optimal policy parameters and system cost in heavy traffic limit, the first of their kind to our knowledge. These expressions sharpen our understanding of the key determinants of the optimal policy and their quantitative and qualitative impacts. For example, the results demonstrate that the well-known square-root relationship between the optimal order quantity and demand rate under a sequential processing environment is replaced by the cube root under a stochastic parallel processing environment. We further extend the study to periodic-review (S,T) systems with constant leadtimes.

Palm distributions are basic tools when studying stationarity in the context of point processes, queueing systems, fluid queues or random measures. The framework varies with the random phenomenon of interest, but usually a one-dimensional group of measure-preserving shifts is the starting point. In the present paper, by alternatively using a framework involving random time changes (RTCs) and a two-dimensional family of shifts, we are able to characterize all of the above systems in a single framework. Moreover, this leads to what we call the detailed Palm distribution (DPD) which is stationary with respect to a certain group of shifts. The DPD has a very natural interpretation as the distribution seen at a randomly chosen position on the extended graph of the RTC, and satisfies a general duality criterion: the DPD of the DPD gives the underlying probability P in return.To illustrate the generality of our approach, we show that classical Palm theory for random measures is included in our RTC framework. We also consider the important special case of marked point processes with batches. We illustrate how our approach naturally allows one to distinguish between the marks within a batch while retaining nice stationarity properties.

We establish heavy-traffic limits for "nearly deterministic" queues, such as the G/D/n many-server queue. Waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated Gn/D/ 1 model, where the G n denotes "cyclic thinning" of order n , indicating that the original (possibly general) point process of arrivals is thinned to contain only every n th point. We thus focus on the Gn/D/ 1 model and the generalization to Gn/Gn/ 1, where "cyclic thinning" is applied to both the arrival and service processes. As n ↓ ∞, the Gn/Gn/ 1 models approach the deterministic D/D/ 1 model. The classical example is the Erlang En/En/ 1 queue, where cyclic thinning of order n is applied to both the interarrival times and the service times, starting from a "base" M/M/ 1 model. We establish different limits in two cases: (i) when (1--ρ n )√ n ↓ β as n ↓ ∞ and (ii) (1 -- ρ n ) n √ as n ↓ ∞, where ρ n is the traffic intensity in model n , and 0 < β < ∞. The nearly deterministic feature leads to interesting nonstandard scaling. We also establish revealing heavy-traffic limits for the stationary waiting times and other performance measures in the Gn/Gn/ 1 queues, by letting ρ n ↑ 1 as n ↓∞.

We study the tail asymptotics of the r.v. X(T) where {X(t)} is a stochastic process with a linear drift and satisfying some regularity conditions like a central limit theorem and a large deviations principle, and T is an independent r.v. with a subexponential distribution. We find that the tail of X(T) is sensitive to whether or not T has a heavier or lighter tail than a Weibull distribution with tail e−x. This leads to two distinct cases, heavy tailed and moderately heavy tailed, but also some results for the classical light-tailed case are given. The results are applied via distributional Little’s law to establish tail asymptotics for steady-state queue length in GI/GI/1 queues with subexponential service times. Further applications are given for queues with vacations, and M/G/1 busy periods.

Consider a stable FIFO GI/GI/1\,\rightarrow\,/GI/1 tandem queue in which the equilibrium distribution of service time at the second node S(2) is subexponential. It is shown that when the service time at the first node has a lighter tail, the tail of steady-state delay at the second node, D(2), has the same asymptotics as if it were a GI/GI/1 queue: P(D(2) > x) \sim {{\rho_2} \over {1-\rho_2}}P(S_{\mathrm{e}}(2) >x), \quad x{\rightarrow}{\infty}, where S_{\mathrm{e}}(2) has equilibrium (integrated tail) density P(S(2) >x)/ {E[S(2)]}, and \rho_2 = \lambda {E[S(2)]} (\lambda is the arrival rate of customers). The same result holds for tandem queues with more than two stations. For split-match (fork-join) queues with subexponential service times, we derive the asymptotics for both the sojourn time and the queue length. Finally, more generally, we consider feedforward generalized Jackson networks and obtain similar results.

Join-the-Shortest-Queue (JSQ) is a very old and popular routing policy for server farms. Figure 1 shows two examples of server farm architectures employing JSQ routing. In both cases, each incoming job is immediately dispatched, via a front-end router, to the queue with the fewest number of jobs, designated as the shortest queue (ties are broken at random). In Figure 1(a), jobs at a queue are served in First-Come-First-Served (FCFS) order. In Figure 1(b), jobs within a queue are served according to Processor-Sharing (PS), meaning that when there are n jobs at a queue, they share the processing capacity, each simultaneously receiving 1/nth of the service. We refer to Figure 1(a) as a JSQ/FCFS server farm and to Figure 1(b) as a JSQ/PS farm. If more detail is needed, we use the notation: M/G/K/JSQ/PS, denoting a Poisson arrival process, i.i.d. job sizes from a general distribution, K servers, JSQ routing; and PS scheduling at queues.

In Mandelbaum and Yechiali [The conditional residual service time in the M/G/1 queue, http://www.math.tau.ac.il/~uriy/publications (No. 30a), 1979] and in Fakinos [The expected remaining service time in a single-server queue, Oper. Res. 30 (1982) 1014-1018] a simple formula is derived for the (stationary) expected remaining service time in a M/G/1 queue, conditional on the number of customers in the system. We give a short new proof of the formula using Rate Conservation Law, and generalize to handle higher moments.

We investigate the question of when sampling a stochastic process X={X(t) : t⩾0} at the times of an independent point process ψ leads to the same empirical distribution as the time-average limiting distribution of X. Two main cases are considered. The first is when X is asymptotically stationary and ergodic, and ψ satisfies a mixing condition. In this case, the pathwise limiting distributions in function space are shown to be the same. The second main case is when X is only assumed to have a constant finite time average and ψ is assumed a positive recurrent renewal processes with a spread-out cycle length distribution. In this latter case, the averages are shown to be the same when some further conditions are placed on X and ψ.

Most bounds for expected delay, E[D], in GI/GI/c queues are modifications of bounds for the GI/GI/1 case. In this paper we exploit a new delay recursion for the GI/GI/c queue to produce bounds of a different sort when the traffic intensity p = λ/μ = E[S]/E[T] is less than the integer portion of the number of servers divided by two. (S AND T denote generic service and interarrival times, respectively.) We derive two different families of new bounds for expected delay, both in terms of moments of S AND T. Our first bound is applicable when E[S2] < ∞. Our second bound for the first time does not require finite variance of S; it only involves terms of the form E[Sβ], where 1 < β < 2. We conclude by comparing our bounds to the best known bound of this type, as well as values obtained from simulation.