In this paper, we present an extension to the classical SIR epidemic transmission model that uses any general probability distribution for the length of the infectious period. The classical SIR model implicitly requires an exponential distribution for the length of this period of time. We will show how a general distribution can be easily taken into account using the Transient Little Law and present numerical methods to solve the model in an efficient way. Our numerical experiments show that in the presence of a more realistic distribution, with lower variability than the exponential distribution, the size of peak of infected individuals on the graph will be higher and occur earlier. Conversely, a higher-variability distribution will lead to a lower peak that takes longer to dissipate. We also discuss some extensions to the basic model, to include variants like SEIRD and SIS. These findings should have profound and important consequences in the design of public policy.
In recent years, well-designed bus rapid transit (BRT) systems have become a real alternative to more expensive rail-based public transportation systems around the world. However, once the BRT system is operational, its success often depends on the routes offered to passengers. Thus, the bus rapid transit route design problem (BRTRDP) is the problem of finding a set of routes and frequencies that minimizes the operational and passenger costs (travel time) while simultaneously satisfying the system’s technical constraints, such as meeting the demands for trips, bus frequencies, and lane capacities. To address this problem, we propose a mathematical formulation of the BRTRDP as a mixed-integer program (MIP) with an underlying network structure. However, because of the vast number of routes, solving the MIP via branch and bound is out of reach for most practical instances. Hence, we propose a decomposition strategy that, given a certain set of routes, decouples the route selection decisions from the BRT system performance evaluation. The latter evaluation is done by solving a linear optimization problem using a column generation scheme. We embedded this decomposition strategy in a hybrid genetic algorithm (HGA) and tested it in 14 instances ranging from 5 to 40 stations with different BRT system topologies. The results show that in 8 of 14 problems, the HGA was able to obtain a solution that is provably optimal within 0.20%. Additionally, in 4 of 14 instances, HGA obtained the optimal solution.
We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we show that previous laws (see Bertsimas and Mourtzinou (1997)) can be generalized. Furthermore, within this framework, a new law can be derived as well, which gives higher-moment expressions for very general types of queueing system; in particular, the laws hold for systems that allow customers to overtake one another. What is especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide expressions for all moments of the number of customers in the system in an M/G/1 preemptive last-come-first-served queue at a time t > 0, for any initial condition and any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptiverepeat with and without resampling) that are analogous to the special cases found in Whitt (1987c), (1988). These expressions are then used to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see Whitt (1987a), (1987b)).
This paper describes linear programming solvers for Markov Decision Processes, as an extension to the JMDP program. JMDP is an object-oriented framework to model and solve Markov Decision Processes (MDP) programmed in Java. The developed solvers work for the Discounted Cost and Average Cost criteria. Our solvers are compared with existing Value Iteration and Policy Iteration solvers.
We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we show that previous laws (see Bertsimas and Mourtzinou (1997)) can be generalized. Furthermore, within this framework, a new law can be derived as well, which gives higher-moment expressions for very general types of queueing system; in particular, the laws hold for systems that allow customers to overtake one another. What is especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide expressions for all moments of the number of customers in the system in an M/G/1 preemptive last-come-first-served queue at a time t > 0, for any initial condition and any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptive-repeat with and without resampling) that are analogous to the special cases found in Abate and Whitt (1987c), (1988). These expressions are then used to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see Abate and Whitt (1987a), (1987b)).
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