Dynamic Optimization with Applications to Dynamic Rate Queues

Abstract

This tutorial presents recent developments in the management of communications services and applies broadly to services involving the leasing of shared resources. These problems are more realistically modeled by queues with time-varying rates or more simply, dynamic rate queues. We first provide a review and summary of relevant results for various fundamental dynamic rate queues. The focus here is on approximations of these queueing models by low-dimensional dynamical systems. The dynamic optimization of constrained dynamical systems is based on the calculus of variations and its various incarnations over the past three centuries. We discuss these methods in the context of Lagrangians, Hamiltonians, and Bellman value functions. Finally, we provide examples where we apply these optimization techniques to dynamic rate queues motivated by communications decision problems.Keywords Bellman value function; calculus of variations; conservation principles; differential equations; dynamical systems; Hamiltonian; Lagrangian; Lagrange multipliers; Legendre transforms; opportunity costs; optimal control; Poisson brackets The Operations of Communication Systems and Services Four Canonical IssuesWe begin by discussing the operational issues motivated by the communications industry. This is the business sector that created the telephone network in the beginning of the 20th century and gave birth to the mathematical field of queueing theory. Around the middle of the 20th century, or the 1960s, queueing theory was found to have applications to the communication of computer systems. The mathematics of queueing has taken on a life of its own since then and has applications to all services involving the leasing of shared resources. A manager of a communications business has a continual interest in offering various services as efficiently as possible in terms of the company resources. We can express these concerns in terms of four canonical issues: performance, pricing, provisioning, and prioritization. The first one is that of service performance. These are metrics that predict the availability of resources needed for service to the typical, newly arriving customer. This is expressed in terms of the past demand and usage of these resources by previous customers.The second issue is service pricing. If the price is too high, then no one may use the service. If the price is too low, then the cost of the resources used by the business providing the service may exceed the revenue obtained. A decision must be made as to what is the most strategic price in terms of optimizing revenue without compromising some target level of service performance. If the general strategy for a decision can be expressed in terms of a finite set of simple steps, then we call this a policy. Since the larger goal of operations 208

Optimal staffing in nonstationary service centers with constraints

“…Such quality control may be formulated as follows: where and is the maximum allowable probability of abandonment. This quality control is basically an isoperimetric SLA constraint that specifies that during the planning period , the number of customers that abandon, , must be less or equal to the maximum allowable fraction of abandonments . A complete optimal control problem is presented next.…”

“…For a profit center, , as indicated in Theorem 3.1. From the general theory of optimal control, is viewed as the shadow price of one additional service unit or simply the marginal cost rate of one additional server (e.g., see ). This means that is the difference of the revenue r , from served customers, and the marginal cost , for the rendered service.…”

Performance analysis of time-dependent queueing systems: Survey and classification

“…Surveys on timedependent queueing models that are used for staffing decisions in service systems are provided by Green et al [89], Whitt [229], and Defraeye and Van Nieuwenhuyse [56,57]. Hampshire and Massey [96] integrate the performance analysis of time-dependent queueing systems in the optimization of multiple aspects of the communications industry. The applications can be categorized into the areas of telephone call centers, health care facilities, emergency services, service counters, and repair facilities.…”

Development of the dynamic tensor model for traffic management in a telecommunication network with the support of different classes of service

“…Papers [11][12][13][14][15] demonstrated the possibilities of description of dynamics of change in the average queue length on the interfaces with the help of nonlinear differential equations, the form of which is in many respects determined by the nature of the selected models of packets flows and their service discipline on the routers interfaces. At present, the following types of mathematical models, based on different approximations of the dynamics of changes in the state of the network router interface, are known [12] In this enumeration of models, it is worth distinguishing the PSFFA approximation, unquestionable advantage of which is the possibility of assigning dynamics of change in average queue length in the analytical form, which makes it possible to numerically estimate the over time changes of such important parameters of Quality of Service as average delay, jitter of packets, as well as a probability of the packets loss.…”