Bicriteria shortest path in networks of queues
Introduction
Due to the vast applications of network of queues, it is one of the most important issues in queueing theory. The complexity of analyzing this subject is another reason that some researchers are still investigating it from the various points of view.
Many problems in the fields of transportation networks, production systems, computer networks and distributed processing systems are expressed in the framework of networks of queues. For example, consider the problem of routing a ship in the ocean. In this case, nodes of the network of queues indicate the geographical regions where the ship passes through and the ship should wait in each region (service station) for anchoring and fuelling. The transport times from each region to the adjacent regions are assumed to be mutually independent random variables with general distributions. The length of each path in the network is equal to the sum of the lengths of the arcs and the nodes of this path. The length of each node is equal to the waiting time for anchoring and fuelling in the particular region (waiting time in queue plus the processing time), and the arc lengths are equal to the transport times between the service stations. In each region, the number of servers is either one or infinity. In practice, a queueing system with infinite servers indicates that there is ample capacity so that no ship ever has to wait. Finally, we can obtain the shortest path length from the origin to the destination in this ship routing problem, considering the mean length and also its variance, as two important criteria in routing problems, using the proposed method.
Azaron and Kianfar [1] proposed an exponential algorithm to find the dynamic shortest path in the ship routing problem, but the waiting time in each region for anchoring and fuelling, which is an important factor to compute the shortest path length, has not been considered in this paper. The proposed algorithm in this paper can overcome this drawback.
In the proposed methodology, if a service station exists in any node of the network, then a proper stochastic arc corresponding to the waiting time in the particular service station is augmented to the network. Then, the length of any arc in the transformed stochastic network is replaced by two deterministic criteria corresponding to the mean and the variance of the arc length. Finally, the best bicriteria path is found by using dynamic programming, considering a utility function, which includes two indicated criteria.
There are several papers, which obtain a specific path as the shortest path in stochastic networks, considering different criteria like minimum expected path length, minimum of the variance of the path length or the maximum probability that the shortest path length becomes smaller than a specific value. Lee and Pulat [9] presented a method for solving the bicriteria network flows problems. The network flows include a wide range of problems like transportation, maximal flow and shortest path. The efficient extreme points of the bicriteria network are found by a combination of the bicriteria linear programming with the out-of-kilter method. Since the decision variables of the shortest path problem are zero-one, this algorithm does not have enough efficiency for solving the shortest path problem. The main approach for solving the multicriteria shortest path problems is to obtain the efficient paths of the network. For example, Martins [10] found the set of efficient paths for the bicriteria shortest path problem by using dynamic programming. Current and Min [6] provided a taxonomy and annotation for the multicriteria networks. Wijerante et al. [11] presented a method for finding the set of non-dominated paths from the source to the sink node. They assumed that each arc includes several criteria and some of them might be stochastic. They replaced each stochastic criterion with two deterministic criteria corresponding to the mean and the variance of the stochastic criterion. Then they found the set of non-dominated paths by a multicriteria algorithm. Carraway et al. [5] applied the method of generalized dynamic programming for finding the shortest path of a bicriteria network, in which one criterion is corresponding to the shortest path and the other criterion is corresponding to the most reliable path of the network. They proved that DP’s monotonicity assumption is violated in this situation. Therefore, they used generalized DP, which avoids the potential pitfalls created by this absence of monotonicity, thereby guaranteeing optimality.
Although there are several papers corresponding to the multicriteria networks and also the multicriteria shortest path problems, but there is no paper about finding the multicriteria shortest path in networks of queues and its applications in different fields like stochastic routing problems and also stochastic dynamic scheduling problems.
Section snippets
Shortest path analysis in networks of queues
In this section, first I propose a method for transforming an acyclic network of queues (called original network) into an equivalent stochastic network, by replacing each node containing a queueing system with a stochastic arc. The length of this arc is equal to the waiting time in the queueing system located in the corresponding node of the original network. Then, the stochastic network is transformed into a bicriteria network by computing the mean and the variance of the waiting time in each
Bicriteria path
A path π is a sequence (I1, … , In) of two or more nodes (n ⩾ 2), in which (Ik, Ik+1) ∈ A for k = 1, 2, … , n − 1. Let P represent the set of all paths of the network and p(j) = {π ∈ P/I1 = 0, In = j} represent the set of all paths ending to node j from the source node.
Assume that for each arc (i, j) ∈ A there is the value vector , in which μij represents the mean and represents the variance of the length of the arc. I define the path value function z : P → R2 in this manner: , in
Dynamic programming approach
The application of dynamic programming for solving multiple criteria problems with a specific utility function were considered in several papers. If we can prove that the utility function satisfies the monotonicity assumption, we can easily use dynamic programming for solving the problem (see Bellman [4] for more details).
Unfortunately, when we encounter a multicriteria utility function, the proof is not easy. We can extend the method of forward single criterion dynamic programming for
Numerical example
Consider the network of queues shown in Fig. 1.
The assumptions are as follows:
- 1.
There is only one external arrival process to node 1 according to a Poisson process with the average of 0.8 per hour (λ = 0.8).
- 2.
There is a service station with infinite servers in node 1 whose distribution of service time is normal with parameters (μ, σ2) = (10, 1)(B′ (t) = N(10, 1)).
- 3.
There is a service station with one server in node 2 whose distribution of service time is gamma with parameters (α, β = 2, 1) (B′(t) = te−t t > 0).
- 4.
There
Conclusion
In this paper, I developed a polynomial algorithm to find the shortest path from the source to the sink node of a network of queues in the steady-state, based on queueing theory, dynamic programming, graph theory and also multiple criteria decision making. It was assumed that some nodes of the network contain service stations including either one or infinite servers with general distributions of service time. Moreover, the arc lengths among the service stations are assumed to be independent
Acknowledgements
This research is supported by Science Foundation Ireland under Grant No. 03/CE3/I405 as part of the Centre for Telecommunications Value-Chain-Driven Research (CTVR) and under Grant No. 00/PI.1/C075.
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