A basic calculus is presented for stochastic service guarantee analysis in communication networks. Central to the calculus are two definitions, maximum-(virtual)-backlog-centric (m.b.c) stochastic arrival curve and stochastic service curve, which respectively generalize arrival curve and service curve in the deterministic network calculus framework. With m.b.c stochastic arrival curve and stochastic service curve, various basic results are derived under the (min, +) algebra for the general case analysis, which are crucial to the development of stochastic network calculus. These results include (i) superposition of flows, (ii) concatenation of servers, (iii) output characterization, (iv) per-flow service under aggregation, and (v) stochastic backlog and delay guarantees. In addition, to perform independent case analysis, stochastic strict server is defined, which uses an ideal service process and an impairment process to characterize a server. The concept of stochastic strict server not only allows us to improve the basic results (i) -- (v) under the independent case, but also provides a convenient way to find the stochastic service curve of a serve. Moreover, an approach is introduced to find the m.b.c stochastic arrival curve of a flow and the stochastic service curve of a server.
One type uses a sequence of random variables to stochastically bound the arrival process @cite_34 or the service process @cite_38 . Similar properties as (P.1), (P.3), (P.4) and (P.5) have been studied @cite_34 @cite_38 . These studies generally need the independence assumption. Under this type of traffic and service models, several problems remain open, which are out of the scope of this paper. One is the concatenation property (P.2), another is the general case analysis and the third is researching designing approaches to map known traffic and service characterizations to the required sequences of random variables.
@cite_38 @cite_34
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The problem of transporting patients or elderly people has been widely studied in literature and is usually modeled as a dial-a-ride problem (DARP). In this paper we analyze the corresponding problem arising in the daily operation of the Austrian Red Cross. This nongovernmental organization is the largest organization performing patient transportation in Austria. The aim is to design vehicle routes to serve partially dynamic transportation requests using a fixed vehicle fleet. Each request requires transportation from a patient's home location to a hospital (outbound request) or back home from the hospital (inbound request). Some of these requests are known in advance. Some requests are dynamic in the sense that they appear during the day without any prior information. Finally, some inbound requests are stochastic. More precisely, with a certain probability each outbound request causes a corresponding inbound request on the same day. Some stochastic information about these return transports is available from historical data. The purpose of this study is to investigate, whether using this information in designing the routes has a significant positive effect on the solution quality. The problem is modeled as a dynamic stochastic dial-a-ride problem with expected return transports. We propose four different modifications of metaheuristic solution approaches for this problem. In detail, we test dynamic versions of variable neighborhood search (VNS) and stochastic VNS (S-VNS) as well as modified versions of the multiple plan approach (MPA) and the multiple scenario approach (MSA). Tests are performed using 12 sets of test instances based on a real road network. Various demand scenarios are generated based on the available real data. Results show that using the stochastic information on return transports leads to average improvements of around 15 . Moreover, improvements of up to 41 can be achieved for some test instances. In 1991, D. J. Bertsimas and G. van Ryzin introduced and analyzed a model for stochastic and dynamic vehicle routing in which a single, uncapacitated vehicle traveling at a constant velocity in a Euclidean region must service demands whose time of arrival, location and on-site service are stochastic. The objective is to find a policy to service demands over an infinite horizon that minimizes the expected system time (wait plus service) of the demands. This paper extends our analysis in several directions. First, we analyze the problem of m identical vehicles with unlimited capacity and show that in heavy traffic the system time is reduced by a factor of 1 m2 over the single-server case. One of these policies improves by a factor of two on the best known system time for the single-server case. We then consider the case in which each vehicle can serve at most q customers before returning to a depot. We show that the stability condition in this case depends strongly on the geometry of the region. Several pol... This paper contains a quantitative evaluation of probabilistic traffic assignment models and proposes an alternate formulation. First, the concept of stochastic-user-equilibration (S-U-E) is formalized as an extension of Wardrop's user-equilibration criterion. Then, the stochastic-network-loading (S-N-L) problem (a special case of S-U-E for networks with constant link costs) is analyzed in detail and an expression for the probability of route choice which is based on two general postulates of user behavior is derived. The paper also discusses the weaknesses of existing S-N-L techniques with special attention paid to Dial's multipath method and compares them to the suggested approach. The proposed model seems reasonable and does not exhibit the inherent weaknesses of the logit model when applied to sets of routes which overlap heavily. The discussion is supported by several numerical examples on small contrived networks. The paper concludes with the discussion of two techniques that can be used to approximate the link flows resulting from the proposed model in large networks. In this paper, we undertake the first study of statistical multiplexing from the perspective of approximation algorithms. The basic issue underlying statistical multiplexing is the following: in high-speed networks, individual connections (i.e., communication sessions) are very bursty, with transmission rates that vary greatly over time. As such, the problem of packing multiple connections together on a link becomes more subtle than in the case when each connection is assumed to have a fixed demand. We consider one of the most commonly studied models in this domain: that of two communicating nodes connected by a set of parallel edges, where the rate of each connection between them is a random variable. We consider three related problems: (1) stochastic load balancing, (2) stochastic bin-packing, and (3) stochastic knapsack. In the first problem the number of links is given and we want to minimize the expected value of the maximum load. In the other two problems the link capacity and an allowed overflow probability p are given, and the objective is to assign connections to links, so that the probability that the load of a link exceeds the link capacity is at most @math . In bin-packing we need to assign each connection to a link using as few links as possible. In the knapsack problem each connection has a value, and we have only one link. The problem is to accept as many connections as possible. For the stochastic load balancing problem we give an O(1)-approximation algorithm for arbitrary random variables. For the other two problems we have algorithms restricted to on-off sources (the most common special case studied in the statistical multiplexing literature), with a somewhat weaker range of performance guarantees. A standard approach that has emerged for dealing with probabilistic resource requirements is the notion of effective bandwidth---this is a means of associating a fixed demand with a bursty connection that "represents" its distribution as closely as possible. Our approximation algorithms make use of the standard definition of effective bandwidth and also a new one that we introduce; the performance guarantees are based on new results showing that a combination of these measures can be used to provide bounds on the optimal solution.