# American Institute of Mathematical Sciences

ISSN:
1547-5816

eISSN:
1553-166X

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## Journal of Industrial & Management Optimization

July 2010 , Volume 6 , Issue 3

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2010, 6(3): 453-464 doi: 10.3934/jimo.2010.6.453 +[Abstract](1372) +[PDF](135.2KB)
Abstract:
We formulate a network equilibrium model with a unilateral constraint based on Wardrop's principle and establish that the model is equivalent to a quasi-variational inequality. For the case with vector cost functions, we use Gerstewitz's function to derive two equivalence relations between the vector network equilibrium model with a unilateral constraint and quasi-variational inequalities without any convexity assumptions.
2010, 6(3): 465-481 doi: 10.3934/jimo.2010.6.465 +[Abstract](1368) +[PDF](216.1KB)
Abstract:
In this paper, we study the Levitin-Polyak type well-posedness of variational inequalities and optimization problems with variational inequality constraints in Banach spaces. We derive some criteria and characterizations for these Levitin-Polyak well-posedness. We also investigate conditions under which the existence and uniqueness of solution is equivalent to the Levitin-Polyak well-posedness of the problem.
2010, 6(3): 483-496 doi: 10.3934/jimo.2010.6.483 +[Abstract](1851) +[PDF](202.0KB)
Abstract:
This paper investigates a benchmark and a mean-variance portfolio selection problems for insurers under the model assumptions of Yang and Zhang [20]. Closed-form expressions for the value functions, the optimal investment strategies and the mean-variance efficient frontier are achieved by using the stochastic maximum principle. The optimal strategies are expressed directly in terms of the insurer's wealth process and hence can be easily applied in practice. And a numerical example is given to illustrate our results.
2010, 6(3): 497-500 doi: 10.3934/jimo.2010.6.497 +[Abstract](1309) +[PDF](86.4KB)
Abstract:
In this note, we establish a mixed type converse duality for a class of multiobjective programming programs. This clarifies several omissions in an earlier work by Yang et al. [Mixed type converse duality in multiobjective programming problems, J. Math. Anal. Appl. 304 (2005) 394-398].
2010, 6(3): 501-516 doi: 10.3934/jimo.2010.6.501 +[Abstract](1189) +[PDF](192.9KB)
Abstract:
In this paper, we study a two-server Markovian network system with balking and a Bernoulli schedule under a single vacation policy, where servers have different service rates. After every service, only one server may take a vacation or continue to stay in the system. The vacation time follows an exponential distribution. An arriving customer finding both servers free will choose the faster server. If the customer finds only one server is free, this customer chooses this free server. If the customer finds both servers are not free, then this customer may join the system or balk. For this system, we obtain the steady state condition, the stationary distribution of the number of customers in the system, and the mean system size by using a matrix-geometric method. Some special cases are deduced, which match with earlier exiting results. Extensive numerical illustrations are provided. Motivation for this system model also comes from some computer communication networks with different types of traffic such as real-time traffic and non-real-time traffic, where messages can be processed by two channels (servers) with different transmission rates. The behavior of abandoning messages can be equated with the balking of customers in this system model.
2010, 6(3): 517-540 doi: 10.3934/jimo.2010.6.517 +[Abstract](1528) +[PDF](358.9KB)
Abstract:
This paper considers a state-dependent M/M/$c$/$c+r$ retrial queue with Bernoulli abandonment, where the number of servers is equal to $c$, the capacity of the buffer is equal to $r$ and that of the virtual waiting room (called orbit) for the retrial customers is infinite. We assume that the arrival, service and retrial rates depend on the number of customers in the system (the servers and buffer). In this paper, we first present the ergodic condition for our retrial queue. Then, by a continued fraction approach, we derive analytical solutions for the stationary joint distribution of the queue lengths in the system and in the orbit, assuming that the capacity of the system is less than or equal to 4. We further show that our analytical solutions can be computed with any desired accuracy. Finally, we present some numerical results to show the impact of the parameters on the performance of the system.
2010, 6(3): 541-568 doi: 10.3934/jimo.2010.6.541 +[Abstract](1319) +[PDF](1740.4KB)
Abstract:
In this paper, we investigate mean sojourn times in multiclass queues with feedback and their application to packet scheduling in communication networks.
We first analyze the feedback queues with $J$ groups of customers and a single server. The server selects a group (station) in a priority order and admits its customers into the service facility according to one of the following rules: 1-limited, gated, or exhaustive. Since there are multiple classes of customers in each group, they are served in either the (local) priority order or the first-come-first-served order. We derive the mean sojourn times for all service stages of customers circulating the system.
Then we consider their application to a packet scheduling problem in communication networks where a mission-critical traffic is given some preferential treatment over other normal traffic. Packets belonging to normal traffic are allocated the resource capacity in proportion to their assigned weights in a round-robin manner, which we call the "discriminatory round robin (DRR)." We construct some composite scheduling algorithms and compare their performance.
2010, 6(3): 569-585 doi: 10.3934/jimo.2010.6.569 +[Abstract](1416) +[PDF](242.7KB)
Abstract:
Providing a photonic alternative to the current electronic switching in the backbone, optical packet switching (OPS) and optical burst switching (OBS) require optical buffering. Optical buffering exploits delays in long optical fibers; an optical buffer is implemented by routing packets through a set of fiber delay lines (FDLs). Previous studies pointed out that, in comparison with electronic buffers, optical buffering suffers from an additional performance degradation. This contribution builds on this observation by studying optical buffer performance under more general traffic assumptions. Features of the optical buffer model under consideration include a Markovian arrival process, general burst sizes and a finite set of fiber delay lines of arbitrary length. Our algorithmic approach yields instant analytic results for important performance measures such as the burst loss ratio and the mean delay.
2010, 6(3): 587-602 doi: 10.3934/jimo.2010.6.587 +[Abstract](1317) +[PDF](203.1KB)
Abstract:
This paper studies a single-server non-preemptive priority queue with two traffic classes in order to model Expedited Forwarding Per-Hop Behavior in the Differentiated Services (DiffServ) architecture. Generally, queueing models assume infinite queue capacity but in a DiffServ router the capacity for high priority traffic is typically small to prevent this traffic from monopolizing the output link and hence causing starvation of low-priority traffic. The presented model takes the exact (finite) high-priority queue capacity into account. Analytical formulas for the system content of each class are determined as well as the high-priority packet loss ratio. For each class, service of a packet takes a (different) general independent distribution. The issues this causes are resolved by using spectral decomposition. Numerical examples indicate the considerable impact of the finite capacity on system performance.
2010, 6(3): 603-619 doi: 10.3934/jimo.2010.6.603 +[Abstract](1163) +[PDF](248.4KB)
Abstract:
In this paper, we formulate and analyze a model of the resequencing buffer at the receiver's side for the Selective Repeat protocol over a general class of transmission channels. Thanks to its efficiency, Selective Repeat is a ubiquitous error control mechanism in many different settings, in particular in wireless protocols such as WiMax and WiFi.
In view of the correlated nature of transmission errors over wireless channels, the receiver buffer model considers a general Markovian error process. We provide both an exact mathematical analysis of the receiver buffer behavior as well as a computationally efficient large-deviations result. An asymptotic analysis of the delay is also given. Numerical examples show that the correlation of the error process has an important influence on the performance of the receiver buffer.
2010, 6(3): 621-639 doi: 10.3934/jimo.2010.6.621 +[Abstract](1419) +[PDF](302.9KB)
Abstract:
In wireless networks, guard channel based Call Admission Control (CAC) schemes are widely used to protect handoff calls for seamless wireless services. However, the introduction of guard channels in wireless networks results in the increase in the new call blocking probability. This general situation also occurs in a 3G-WLAN integrated network. To solve this problem in a 3G-WLAN integrated network for QoS support, we propose an adaptive guard channel based CAC scheme in this paper. The main objective of our guard channel based CAC scheme is to alleviate the increase in the new call blocking probability due to guard channels while it can still protect handoff calls. For the design of our adaptive guard channel based CAC scheme, we first develop and analyze a performance model for a general guard channel based CAC scheme based on a level-dependent quasi birth and death model. The analytic results are used to design our adaptive guard channel based CAC scheme. Numerical studies show that our adaptive guard channel based CAC scheme can achieve its objective and its performance is near optimal.
2010, 6(3): 641-660 doi: 10.3934/jimo.2010.6.641 +[Abstract](1419) +[PDF](344.4KB)
Abstract:
2010, 6(3): 661-690 doi: 10.3934/jimo.2010.6.661 +[Abstract](1375) +[PDF](342.4KB)
Abstract:
The paper deals with the continuous-time BM AP/G/1 queue with multiple vacations and with its application to IEEE 802.16e sleep mode. The lengths of the vacation periods have general distribution and they depend on the number of preceding vacations (dependent multiple vacation). We give the expressions for the vector generating function of the stationary number of customers and its mean. Moreover we obtain new formulas for the vector Laplace-Stieljes transform of the stationary virtual waiting time and for its first two moments in case of First-Come First-Serve scheduling.
We apply this vacation model to the IEEE 802.16e sleep mode mechanism, and we evaluate its performance as a function of the traffic intensity and the traffic correlation parameter. We give an example for determining the best sleep mode parameters for a simple optimization criteria and we also develop a cost model for the more general case. For traffic modeling we use a two-phase Markovian Arrival Process, which is appropriate to model a fairly general correlated traffic.
2010, 6(3): 691-708 doi: 10.3934/jimo.2010.6.691 +[Abstract](1566) +[PDF](238.2KB)
Abstract:
In this paper, we present an enhanced power saving class (PSC) type III by introducing a sleep-delay mechanism into the conventional PSC type III in the IEEE 802.16e network. Two multiple vacation queueing models: without sleep-delay and with sleep-delay, are built to capture the working principles of the conventional PSC type III and the enhanced PSC type III, respectively. Then, using the boundary state variable theory method, we analyze these queueing models and give the performance measures in terms of the switch ratio, the power saving ratio, the average response time and the system utility for both the conventional and the enhanced power saving schemes. Next, numerical results are given to demonstrate the dependency relationships between the performance measures and the sleep window size. Moreover, we compare the performance of the both systems without sleep-delay (the conventional PSC type III) and with sleep-delay (the enhanced PSC type III) to validate the effect of the sleep-delay mechanism in the enhanced PSC type III proposed in this paper. Finally, we also construct a cost function to determine the optimal time length of the sleep-delay timer for minimizing the cost function.

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