Level Crossings in Point Processes Applied to Queues: Single-Server Case

Information about delays can enhance service quality in many industries. Delay information can take many forms, with different degrees of precision. Different levels of information have different effects on customers and so on the overall system. The goal of this research is to explore these effects. We first consider a queue with balking under three levels of delay information: No information, partial information (the system occupancy) and full information (the exact waiting time). We assume Poisson arrivals, independent, exponential service times, and a single server. Customers decide whether to stay or balk based on their expected waiting costs, conditional on the information provided. By comparing the three systems, we identify some important cases where more accurate delay information improves performance. In other cases, however, information can actually hurt the provider or the customers.We then investigate the impacts on the system of different cost functions and weight distributions. Specifically, we compare systems where these parameters are related by various stochastic orders, under different information scenarios. We also explore the relationship between customer characteristics and the value of information. The results here are mostly negative. We find that the value of information need not be greater for less patient or more risk-averse customers. After that, we extend our analysis to systems with phase-type service times. Our analytical and numerical results indicate that the previous conclusions about systems with exponential service times still hold for phase-type service times. We also show that service-time variability degrades the system's performance. At last, we consider two richer models of information: In the first model, an arriving customer learns an interval in which the system occupancy falls. In the second model, each customer's service time is the sum of a geometric number of i.i.d. exponential phases, and an arriving customer learns iv the total number of phases remaining in the system. For each information model, we compare two systems, identical except that one has more precise information. We study the effects of information on performance as seen by the service provider and the customers.v

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“…By a level-crossing argument (Brill and Posner 1977), these quantities uniquely satisfy the integral equation…”

Section: Full Informationmentioning

confidence: 99%

Analysis and Comparison of Queues with Different Levels of Delay Information

2007

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“…The proof of this theorem is detailed in [15]. We will show that the state equation (6) can be obtained from this theorem directly.…”

Section: Level Crossing Of Virtual Waiting Timementioning

confidence: 94%

“…An alternative derivation of the state equation (6) can be conducted by the level crossing of virtual waiting time described in [15].…”

Section: Level Crossing Of Virtual Waiting Timementioning

confidence: 99%

Blocking and Delay Analysis of Single Wavelength Optical Buffer With General Packet Size Distribution

Liu

Lee

Jiang

et al. 2009

J. Lightwave Technol.

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Abstract-Buffers are essential components of any packet switch for resolving contentions among arriving packets. Currently, optical buffers are composed of fiber delay lines (FDL), whose blocking and delay behavior differ drastically from that of conventional RAM at least two-fold: 1) only multiples of discrete time delays can be offered to arriving packets; 2) a packet must be dropped if the maximum delay provided by optical buffer is not sufficient to avoid contention, this property is called balking. As a result, optical buffers only have finite time resolution, which may lead to excess load and prolong the packet delay. In this paper, a novel queueing model of optical buffer is proposed, and the closed-form expressions of blocking probability and mean delay are derived to explore the tradeoff between buffer performance and system parameters, such as the length of the optical buffer, the time granularity of FDLs, and to evaluate the overall impact of packet length distribution on the buffer performance.Index Terms-Blocking and delay performance, optical buffer, optical switching, queueing analysis.

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“…• Example 2: In this second example we set θ = [1, 0.5]; µ = [15,25]; c = [0, 0]; d = [5, 3.2], so that c 1 µ 1 ≥c 2 µ 2 andc 2 µ 2 /θ 2 ≥c 1 µ 1 /θ 1 . As explained in Remark 3, setting c 1 = c 2 = 0 gives a different interpretation of the model: customers will abandon the system when a certain deadline is met before they have attained full service.…”

Section: Performance Analysis For K =mentioning

confidence: 99%

“…An important line of research aims at characterizing the performance and impact of abandonments in systems, we refer to [15,8,14,24,26,2] for single-server models and [13,12,33] for papers dealing with the multi-class case. We also refer to [19] for a recent survey on abandonments in a many-server system.…”

Section: Introductionmentioning

confidence: 99%

Dynamic fluid-based scheduling in a multi-class abandonment queue

Larrañaga

Ayesta

Verloop

2013

Performance Evaluation

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We investigate how to share a common resource among multiple classes of customers in the presence of abandonments. We consider two different models: (1) customers can abandon both while waiting in the queue and while being served, (2) only customers that are in the queue can abandon. Given the complexity of the stochastic optimization problem we propose a fluid model as a deterministic approximation. For the overload case we directly obtain that thecµ/θ rule is optimal. For the underload case we use Pontryagin's Maximum Principle to obtain the optimal solution for two classes of customers; there exists a switching curve that splits the two-dimensional state-space into two regions such that when the number of customers in both classes is sufficiently small the optimal policy follows thecµ-rule and when the number of customers is sufficiently large the optimal policy follows thecµ/θ-rule. The same structure is observed numerically in the optimal policy of the stochastic model for an arbitrary number of classes. Based on this we develop a heuristic and by numerical experiments we evaluate its performance and compare it to several index policies. We observe that the suboptimality gap of our solution is small.

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Level Crossings in Point Processes Applied to Queues: Single-Server Case

Cited by 135 publications

References 9 publications

Analysis and Comparison of Queues with Different Levels of Delay Information

Analysis and Comparison of Queues with Different Levels of Delay Information

Blocking and Delay Analysis of Single Wavelength Optical Buffer With General Packet Size Distribution

Dynamic fluid-based scheduling in a multi-class abandonment queue

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