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Journal articles on the topic 'Queue discipline'

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Published: 4 June 2021

Last updated: 17 February 2022

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1

Yamazaki, Genji, Hirotaka Sakasegawa, and J. George Shanthikumar. "A conservation law for single-server queues and its applications." Journal of Applied Probability 28, no. 1 (March 1991): 198–209. http://dx.doi.org/10.2307/3214750.

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We establish a conservation law for G/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in a G/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in a GI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

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2

Yamazaki, Genji, Hirotaka Sakasegawa, and J. George Shanthikumar. "A conservation law for single-server queues and its applications." Journal of Applied Probability 28, no. 01 (March 1991): 198–209. http://dx.doi.org/10.1017/s0021900200039528.

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We establish a conservation law forG/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in aG/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in aGI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

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3

Hirayama, Tetsuji, Masaaki Kijima, and Shoichi Nishimura. "Further results for dynamic scheduling of multiclass G/G/1 queues." Journal of Applied Probability 26, no. 03 (September 1989): 595–603. http://dx.doi.org/10.1017/s0021900200038183.

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We consider discrete-time dynamic scheduling problems of the following three types ofG/G/1 queue withKdifferent customer classes: (i) aG/DFR/1 queue withKclasses under preemptive resume service discipline, (ii) aG/IFR/1 queue with two classes under preemptive resume service discipline, and (iii) aG/G/1 queue with two classes under non-preemptive service discipline. Interchange arguments are used to show that simple index policies of different type minimize the total holding cost of customers in a finite-horizon scheduling period for the three cases. Our results extend the result for aG/M/1 queue by Buyukkoc et al. (1985) to general queues.

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4

Hirayama, Tetsuji, Masaaki Kijima, and Shoichi Nishimura. "Further results for dynamic scheduling of multiclass G/G/1 queues." Journal of Applied Probability 26, no. 3 (September 1989): 595–603. http://dx.doi.org/10.2307/3214416.

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We consider discrete-time dynamic scheduling problems of the following three types of G/G/1 queue with K different customer classes: (i) a G/DFR/1 queue with K classes under preemptive resume service discipline, (ii) a G/IFR/1 queue with two classes under preemptive resume service discipline, and (iii) a G/G/1 queue with two classes under non-preemptive service discipline. Interchange arguments are used to show that simple index policies of different type minimize the total holding cost of customers in a finite-horizon scheduling period for the three cases. Our results extend the result for a G/M/1 queue by Buyukkoc et al. (1985) to general queues.

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5

De Haan, Roland, Ahmad Al Hanbali, Richard J. Boucherie, and Jan-Kees Van Ommeren. "Transient analysis for exponential time-limited polling models under the preemptive repeat random policy." Advances in Applied Probability 52, no. 1 (March 2020): 32–60. http://dx.doi.org/10.1017/apr.2019.51.

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AbstractPolling systems are queueing systems consisting of multiple queues served by a single server. In this paper we analyze two types of preemptive time-limited polling systems, the so-called pure and exhaustive time-limited disciplines. In particular, we derive a direct relation for the evolution of the joint queue length during the course of a server visit. The analysis of the pure time-limited discipline builds on and extends several known results for the transient analysis of an M/G/1 queue. For the analysis of the exhaustive discipline we derive several new results for the transient analysis of the M/G/1 queue during a busy period. The final expressions for both types of polling systems that we obtain generalize previous results by incorporating customer routeing, generalized service times, batch arrivals, and Markovian polling of the server.

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6

Grishechkin, Sergei. "On a relationship between processor-sharing queues and Crump–Mode–Jagers branching processes." Advances in Applied Probability 24, no. 03 (September 1992): 653–98. http://dx.doi.org/10.1017/s0001867800024459.

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The M/G/1 queue with batch arrivals and a queueing discipline which is a generalization of processor sharing is studied by means of Crump–Mode–Jagers branching processes. A number of theorems are proved, including investigation of heavy traffic and overloaded queues. Most of the results obtained are also new for the M/G/1 queue with processor sharing. By use of a limiting procedure we also derive new results concerning M/G/1 queues with shortest residual processing time discipline.

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7

Grishechkin, Sergei. "On a relationship between processor-sharing queues and Crump–Mode–Jagers branching processes." Advances in Applied Probability 24, no. 3 (September 1992): 653–98. http://dx.doi.org/10.2307/1427484.

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The M/G/1 queue with batch arrivals and a queueing discipline which is a generalization of processor sharing is studied by means of Crump–Mode–Jagers branching processes. A number of theorems are proved, including investigation of heavy traffic and overloaded queues. Most of the results obtained are also new for the M/G/1 queue with processor sharing. By use of a limiting procedure we also derive new results concerning M/G/1 queues with shortest residual processing time discipline.

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8

Torkki, Markus, Miika Linna, Seppo Seitsalo, and Pekka Paavolainen. "HOW TO REPORT AND MONITOR THE PERFORMANCE OF WAITING LIST MANAGEMENT." International Journal of Technology Assessment in Health Care 18, no. 3 (2002): 611–18. http://dx.doi.org/10.1017/s0266462302000442.

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Objectives: Potential problems concerning waiting list management are often monitored using mean waiting times based on empirical samples. However, the appropriateness of mean waiting time as an indicator of access can be questioned if a waiting list is not managed well, e.g., if the queue discipline is violated. This study was performed to find out about the queue discipline in waiting lists for elective surgery to reveal potential discrepancies in waiting list management. Methods: There were 1,774 waiting list patients for hallux valgus or varicose vein surgery or sterilization. The waiting time distributions of patients receiving surgery and of patients still waiting for an operation are presented in column charts. The charts are compared with two model charts. One model chart presents a high queue discipline (first in—first out) and another a poor queue discipline (random) queue. Results: There were significant differences in waiting list management across hospitals and patient categories. Examples of a poor queue discipline were found in queues for hallux valgus and varicose vein operations. Conclusions: A routine waiting list reporting should be used to guarantee the quality of waiting list management and to pinpoint potential problems in access. It is important to monitor not only the number of patients in the waiting list but also the queue discipline and the balance between demand and supply of surgical services. The purpose for this type of reporting is to ensure that the priority setting made at health policy level also works in practise.

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9

Ward, Amy R., and Nicholas Bambos. "On stability of queueing networks with job deadlines." Journal of Applied Probability 40, no. 02 (June 2003): 293–304. http://dx.doi.org/10.1017/s0021900200019318.

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In this paper, we consider a single-server queue with stationary input, where each job joining the queue has an associated deadline. The deadline is a time constraint on job sojourn time and may be finite or infinite. If the job does not complete service before its deadline expires, it abandons the queue and the partial service it may have received up to that point is wasted. When the queue operates under a first-come-first served discipline, we establish conditions under which the actual workload process—that is, the work the server eventually processes—is unstable, weakly stable, and strongly stable. An interesting phenomenon observed is that in a nontrivial portion of the parameter space, the queue is weakly stable, but not strongly stable. We also indicate how our results apply to other nonidling service disciplines. We finally extend the results for a single node to acyclic (feed-forward) networks of queues with either per-queue or network-wide deadlines.

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10

Ward, Amy R., and Nicholas Bambos. "On stability of queueing networks with job deadlines." Journal of Applied Probability 40, no. 2 (June 2003): 293–304. http://dx.doi.org/10.1239/jap/1053003545.

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In this paper, we consider a single-server queue with stationary input, where each job joining the queue has an associated deadline. The deadline is a time constraint on job sojourn time and may be finite or infinite. If the job does not complete service before its deadline expires, it abandons the queue and the partial service it may have received up to that point is wasted. When the queue operates under a first-come-first served discipline, we establish conditions under which the actual workload process—that is, the work the server eventually processes—is unstable, weakly stable, and strongly stable. An interesting phenomenon observed is that in a nontrivial portion of the parameter space, the queue is weakly stable, but not strongly stable. We also indicate how our results apply to other nonidling service disciplines. We finally extend the results for a single node to acyclic (feed-forward) networks of queues with either per-queue or network-wide deadlines.

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11

Sakuma, Yutaka. "ASYMPTOTIC BEHAVIOR FOR MArP/PH/2 QUEUE WITH JOIN THE SHORTEST QUEUE DISCIPLINE." Journal of the Operations Research Society of Japan 54, no. 1 (2011): 46–64. http://dx.doi.org/10.15807/jorsj.54.46.

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12

Miyazawa, Masakiyo, and Genji Yamazaki. "Convex ordering of the attained waiting times in single-server queues and related problems." Journal of Applied Probability 28, no. 2 (June 1991): 433–45. http://dx.doi.org/10.2307/3214878.

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The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

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13

Miyazawa, Masakiyo, and Genji Yamazaki. "Convex ordering of the attained waiting times in single-server queues and related problems." Journal of Applied Probability 28, no. 02 (June 1991): 433–45. http://dx.doi.org/10.1017/s0021900200039802.

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The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

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14

van Ommeren, Jan-Kees, Ahmad Al Hanbali, and Richard J. Boucherie. "Analysis of polling models with a self-ruling server." Queueing Systems 94, no. 1-2 (November 22, 2019): 77–107. http://dx.doi.org/10.1007/s11134-019-09639-6.

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AbstractPolling systems are systems consisting of multiple queues served by a single server. In this paper, we analyze polling systems with a server that is self-ruling, i.e., the server can decide to leave a queue, independent of the queue length and the number of served customers, or stay longer at a queue even if there is no customer waiting in the queue. The server decides during a service whether this is the last service of the visit and to leave the queue afterward, or it is a regular service followed, possibly, by other services. The characteristics of the last service may be different from the other services. For these polling systems, we derive a relation between the joint probability generating functions of the number of customers at the start of a server visit and, respectively, at the end of a server visit. We use these key relations to derive the joint probability generating function of the number of customers and the Laplace transform of the workload in the queues at an arbitrary time. Our analysis in this paper is a generalization of several models including the exponential time-limited model with preemptive-repeat-random service, the exponential time-limited model with non-preemptive service, the gated time-limited model, the Bernoulli time-limited model, the 1-limited discipline, the binomial gated discipline, and the binomial exhaustive discipline. Finally, we apply our results on an example of a new polling discipline, called the 1 + 1 self-ruling server, with Poisson batch arrivals. For this example, we compute numerically the expected sojourn time of an arbitrary customer in the queues.

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15

Miyazawa, Masakiyo, and Ronald W. Wolff. "Symmetric queues with batch departures and their networks." Advances in Applied Probability 28, no. 01 (March 1996): 308–26. http://dx.doi.org/10.1017/s0001867800027385.

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Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length distributions are affected by the distributions of interarrival times, service times and departing batch sizes of customers. Since this is not an easy problem even for single departure models, we first concentrate on single-node queues with a symmetric service discipline, which is known to have nice properties. We start with pre-emptive LIFO, a special case of the symmetric service discipline, and then consider symmetric queues with Poisson arrivals. Stability conditions and stationary queue length distributions are obtained for them, and several stochastic order relations are considered. For the symmetric queues and Poisson arrivals, we also discuss their network. Stability conditions and the stationary joint queue length distribution are obtained for this network.

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16

Miyazawa, Masakiyo, and Ronald W. Wolff. "Symmetric queues with batch departures and their networks." Advances in Applied Probability 28, no. 1 (March 1996): 308–26. http://dx.doi.org/10.2307/1427923.

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Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length distributions are affected by the distributions of interarrival times, service times and departing batch sizes of customers. Since this is not an easy problem even for single departure models, we first concentrate on single-node queues with a symmetric service discipline, which is known to have nice properties. We start with pre-emptive LIFO, a special case of the symmetric service discipline, and then consider symmetric queues with Poisson arrivals. Stability conditions and stationary queue length distributions are obtained for them, and several stochastic order relations are considered. For the symmetric queues and Poisson arrivals, we also discuss their network. Stability conditions and the stationary joint queue length distribution are obtained for this network.

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17

Connor, Stephen B., and Wilfrid S. Kendall. "Perfect simulation of M/G/c queues." Advances in Applied Probability 47, no. 4 (December 2015): 1039–63. http://dx.doi.org/10.1239/aap/1449859799.

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In this paper we describe a perfect simulation algorithm for the stable M/G/c queue. Sigman (2011) showed how to build a dominated coupling-from-the-past algorithm for perfect simulation of the super-stable M/G/c queue operating under first-come-first-served discipline. Sigman's method used a dominating process provided by the corresponding M/G/1 queue (using Wolff's sample path monotonicity, which applies when service durations are coupled in order of initiation of service). The method exploited the fact that the workload process for the M/G/1 queue remains the same under different queueing disciplines, in particular under the processor sharing discipline, for which a dynamic reversibility property holds. We generalise Sigman's construction to the stable case by comparing the M/G/c queue to a copy run under random assignment. This allows us to produce a naïve perfect simulation algorithm based on running the dominating process back to the time it first empties. We also construct a more efficient algorithm that uses sandwiching by lower and upper processes constructed as coupled M/G/c queues started respectively from the empty state and the state of the M/G/c queue under random assignment. A careful analysis shows that appropriate ordering relationships can still be maintained, so long as service durations continue to be coupled in order of initiation of service. We summarise statistical checks of simulation output, and demonstrate that the mean run-time is finite so long as the second moment of the service duration distribution is finite.

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18

Kella, Offer, Bert Zwart, and Onno Boxma. "Some Time-Dependent Properties of Symmetric M/G/1 Queues." Journal of Applied Probability 42, no. 01 (March 2005): 223–34. http://dx.doi.org/10.1017/s0021900200000176.

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We consider an M/G/1 queue that is idle at time 0. The number of customers sampled at an independent exponential time is shown to have the same geometric distribution under the preemptive-resume last-in-first-out and the processor-sharing disciplines. Hence, the marginal distribution of the queue length at any time is identical for both disciplines. We then give a detailed analysis of the time until the first departure for any symmetric queueing discipline. We characterize its distribution and show that it is insensitive to the service discipline. Finally, we study the tail behavior of this distribution.

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Kella, Offer, Bert Zwart, and Onno Boxma. "Some Time-Dependent Properties of Symmetric M/G/1 Queues." Journal of Applied Probability 42, no. 1 (March 2005): 223–34. http://dx.doi.org/10.1239/jap/1110381382.

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We consider an M/G/1 queue that is idle at time 0. The number of customers sampled at an independent exponential time is shown to have the same geometric distribution under the preemptive-resume last-in-first-out and the processor-sharing disciplines. Hence, the marginal distribution of the queue length at any time is identical for both disciplines. We then give a detailed analysis of the time until the first departure for any symmetric queueing discipline. We characterize its distribution and show that it is insensitive to the service discipline. Finally, we study the tail behavior of this distribution.

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20

Ayesta, Urtzi. "A Unifying Conservation Law for Single-Server Queues." Journal of Applied Probability 44, no. 04 (December 2007): 1078–87. http://dx.doi.org/10.1017/s0021900200003752.

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We develop a conservation law for a multi-class GI/GI/1 queue operating under a general work-conserving scheduling discipline. For single-class single-server queues, conservation laws have been obtained for both nonanticipating and anticipating disciplines with general service time distributions. For multi-class single-server queues, conservation laws have been obtained for (i) nonanticipating disciplines with exponential service time distributions and (ii) nonpreemptive nonanticipating disciplines with general service time distributions. The unifying conservation law we develop generalizes already existing conservation laws. In addition, it covers popular nonanticipating multi-class time-sharing disciplines such as discriminatory processor sharing (DPS) and generalized processor sharing (GPS) with general service time distributions. As an application, we show that the unifying conservation law can be used to compare the expected unconditional response time under two scheduling disciplines.

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21

Ayesta, Urtzi. "A Unifying Conservation Law for Single-Server Queues." Journal of Applied Probability 44, no. 4 (December 2007): 1078–87. http://dx.doi.org/10.1239/jap/1197908826.

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We develop a conservation law for a multi-class GI/GI/1 queue operating under a general work-conserving scheduling discipline. For single-class single-server queues, conservation laws have been obtained for both nonanticipating and anticipating disciplines with general service time distributions. For multi-class single-server queues, conservation laws have been obtained for (i) nonanticipating disciplines with exponential service time distributions and (ii) nonpreemptive nonanticipating disciplines with general service time distributions. The unifying conservation law we develop generalizes already existing conservation laws. In addition, it covers popular nonanticipating multi-class time-sharing disciplines such as discriminatory processor sharing (DPS) and generalized processor sharing (GPS) with general service time distributions. As an application, we show that the unifying conservation law can be used to compare the expected unconditional response time under two scheduling disciplines.

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22

Knessl, Charles, and John A. Morrison. "Two Coupled Queues with Vastly Different Arrival Rates: Critical Loading Case." Advances in Operations Research 2011 (2011): 1–26. http://dx.doi.org/10.1155/2011/216790.

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We consider two coupled queues with a generalized processor sharing service discipline. The second queue has a much smaller Poisson arrival rate than the first queue, while the customer service times are of comparable magnitude. The processor sharing server devotes most of its resources to the first queue, except when it is empty. The fraction of resources devoted to the second queue is small, of the same order as the ratio of the arrival rates. We assume that the primary queue is heavily loaded and that the secondary queue is critically loaded. If we let the small arrival rate to the secondary queue beO(ε), where0≤ε≪1, then in this asymptotic limit the number of customers in the first queue will be large, of orderO(ε-1), while that in the second queue will be somewhat smaller, of orderO(ε-1/2). We obtain a two-dimensional diffusion approximation for this model and explicitly solve for the joint steady state probability distribution of the numbers of customers in the two queues. This work complements that in (Morrison, 2010), which the second queue was assumed to be heavily or lightly loaded, leading to mean queue lengths that wereO(ε-1)orO(1), respectively.

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23

Heyman, Daniel P. "The Push-Out-Priority Queue Discipline." Operations Research 33, no. 2 (April 1985): 397–403. http://dx.doi.org/10.1287/opre.33.2.397.

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24

Sengupta, Bhaskar. "A perturbation method for solving some queues with processor sharing discipline." Journal of Applied Probability 26, no. 1 (March 1989): 209–14. http://dx.doi.org/10.2307/3214333.

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In this paper, we present a perturbation method of solving a second-order difference equation with variable coefficients with some additional assumptions. This method can be used to devise an algorithmic solution for the moments of sojourn times in some processor sharing queues. In particular, we examine three queues. The first has exponential service and a fairly general interrupted arrival mechanism. The second is a cyclic queue. The third is a model for a computer system in which finite and infinite sources interact.

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Sengupta, Bhaskar. "A perturbation method for solving some queues with processor sharing discipline." Journal of Applied Probability 26, no. 01 (March 1989): 209–14. http://dx.doi.org/10.1017/s0021900200041966.

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In this paper, we present a perturbation method of solving a second-order difference equation with variable coefficients with some additional assumptions. This method can be used to devise an algorithmic solution for the moments of sojourn times in some processor sharing queues. In particular, we examine three queues. The first has exponential service and a fairly general interrupted arrival mechanism. The second is a cyclic queue. The third is a model for a computer system in which finite and infinite sources interact.

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26

Righter, Rhonda, and J. George Shanthikumar. "Extremal properties of the FIFO discipline in queueing networks." Journal of Applied Probability 29, no. 4 (December 1992): 967–78. http://dx.doi.org/10.2307/3214728.

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We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

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27

Righter, Rhonda, and J. George Shanthikumar. "Extremal properties of the FIFO discipline in queueing networks." Journal of Applied Probability 29, no. 04 (December 1992): 967–78. http://dx.doi.org/10.1017/s0021900200043837.

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We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons: (i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means. (ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means. We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR. Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

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28

Òzekici, Süleyman, Jingwen Li, and Fee Seng Chou. "Waiting Time in M/G/1 Queues with Impolite Arrival Disciplines." Probability in the Engineering and Informational Sciences 9, no. 2 (April 1995): 255–67. http://dx.doi.org/10.1017/s0269964800003843.

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We consider a queueing system where arriving customers join the queue at some random position. This constitutes an impolite arrival discipline because customers do not necessarily go to the end of the line upon arrival. Although mean performance measures like the average waiting time and average number of customers in the queue are the same for all such disciplines, we show that the variance of the waiting time increases as the arrival discipline becomes more impolite, in the sense that a customer is more likely to choose a position closer to the server. For the M/G/1 model, we also provide an iterative procedure for computing the moments of the waiting time distribution. Explicit computational formulas are derived for an interesting special model where a customer joins the queue either at the head or at the end of the line.

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29

Winands, E. M. M., I. J. B. F. Adan, G. J. van Houtum, and D. G. Down. "A STATE-DEPENDENT POLLING MODEL WITH k-LIMITED SERVICE." Probability in the Engineering and Informational Sciences 23, no. 2 (February 16, 2009): 385–408. http://dx.doi.org/10.1017/s0269964809000217.

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We consider a two-queue model with state-dependent setups, in which a single server alternately serves the two queues. The high-priority queue is served exhaustively, whereas the low-priority queue is served according to the k-limited strategy. A setup at a queue is incurred only if there are customers waiting at the polled queue. We obtain the transforms of the queue length and sojourn time distributions under the assumption of Poisson arrivals, generally distributed service times, and generally distributed setup times. The interest for this model is fueled by an application in the field of logistics. It is shown how the results of this analysis can be applied in the evaluation of a stochastic two-item single-capacity production system. From these results we can conclude that significant cost reductions are possible by bounding the production runs of the low-priority item, which indicates the potential of the k-limited service discipline as priority rule in production environments.

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Doytchinov, Bogdan, John Lehoczky, and Steven Shreve. "Real-time queues in heavy traffic with earliest-deadline-first queue discipline." Annals of Applied Probability 11, no. 2 (May 2001): 332–78. http://dx.doi.org/10.1214/aoap/1015345295.

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Sakuma, Yutaka. "Asymptotic behavior for MAP/PH/ queue with shortest queue discipline and jockeying." Operations Research Letters 38, no. 1 (January 2010): 7–10. http://dx.doi.org/10.1016/j.orl.2009.09.011.

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Sakuma, Yutaka, Masakiyo Miyazawa, and Yiqiang Q. Zhao. "Decay rate for a PH/M/2 queue with shortest queue discipline." Queueing Systems 53, no. 4 (August 2006): 189–201. http://dx.doi.org/10.1007/s11134-006-7634-4.

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33

Jain, Gautam, and Karl Sigman. "A Pollaczek–Khintchine formula for M/G/1 queues with disasters." Journal of Applied Probability 33, no. 4 (December 1996): 1191–200. http://dx.doi.org/10.2307/3214996.

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A disaster occurs in a queue when a negative arrival causes all the work (and therefore customers) to leave the system instantaneously. Recent papers have addressed several issues pertaining to queueing networks with negative arrivals under the i.i.d. exponential service times assumption. Here we relax this assumption and derive a Pollaczek–Khintchine-like formula for M/G/1 queues with disasters by making use of the preemptive LIFO discipline. As a byproduct, the stationary distribution of the remaining service time process is obtained for queues operating under this discipline. Finally, as an application, we obtain the Laplace transform of the stationary remaining service time of the customer in service for unstable preemptive LIFO M/G/1 queues.

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Jain, Gautam, and Karl Sigman. "A Pollaczek–Khintchine formula for M/G/1 queues with disasters." Journal of Applied Probability 33, no. 04 (December 1996): 1191–200. http://dx.doi.org/10.1017/s0021900200100580.

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A disaster occurs in a queue when a negative arrival causes all the work (and therefore customers) to leave the system instantaneously. Recent papers have addressed several issues pertaining to queueing networks with negative arrivals under the i.i.d. exponential service times assumption. Here we relax this assumption and derive a Pollaczek–Khintchine-like formula for M/G/1 queues with disasters by making use of the preemptive LIFO discipline. As a byproduct, the stationary distribution of the remaining service time process is obtained for queues operating under this discipline. Finally, as an application, we obtain the Laplace transform of the stationary remaining service time of the customer in service for unstable preemptive LIFO M/G/1 queues.

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35

Indrajaya, Drajat, and Riri Cornellia. "Analisis Model Antrian Loket Transaksi pada PT. POS Indonesia (persero) Kantor Cabang Sawangan dengan Menggunakan Software Promodel." STRING (Satuan Tulisan Riset dan Inovasi Teknologi) 3, no. 2 (December 6, 2018): 170. http://dx.doi.org/10.30998/string.v3i2.2828.

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Simulation is a process of imitation of a tangible thing and its surroundings. The problem found in this study is a long queue at the payment counter resulting from numerous customers who come to POS Company. It is also caused by some factors such as a small number of cash registers, with only 2 counters functioned irregularly and less extensive queue lines. Considering that, an analysis of the queues at the payment counters using pro model software application is made to anticipate the long queues. This simulation is run for 6 hours with 10-time replication, verified and is then validated. The queue model used is (M/M/2):(FCFS/~/~), meaning the number of arrivals uses Poisson distribution and the service time uses exponential distribution. With the number of facility service as many as 2 people, the discipline used is the customer who comes first will be served first (first come first serve). The number of queues in the queue system and the size of the population at the source of arrival is infinite. The average percentage of teller utilities is 62.14% and the idle percentage of teller is 37.86%. The average number of customers completing transactions is 119.30 ? 120 customers for 6 hours. The average time spent by customers in the queue until they are served is 3.67 minutes. The average time customer spends in the system is 6.08 minutes.

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Rosberg, Zvi, and Parviz Kermani. "Customer routing to different servers with complete information." Advances in Applied Probability 21, no. 04 (December 1989): 861–82. http://dx.doi.org/10.1017/s0001867800019091.

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In this paper we consider a queueing system having n exponential servers, each with its own queue and service rate. Customers arrive according to a Poisson process with rate λ, and upon arrival each customer must be routed to some server's queue. No jockeying amongst the queues is allowed and each server serves its queue according to a first-come-first-served discipline. Each server i, 1 ≦ i ≦ n, provides service with a state-dependent rate μ (i)(k), k = 0, 1, …. In addition, at every queue i, there is a deterministic holding cost which occurs at rate h (i)(k) while there are k customers at the queue. An admissible routing policy is a policy that assigns each arriving customer to one of the queues. A decision at time t may be randomized and dependent on the queue lengths and decisions till time t . An optimal routing policy is an admissible policy that minimizes the long-run average holding cost. In this study, we bound the optimal cost from below, by considering an ideal system, where each server optimally selects a given proportion of customers, irrespective of other servers' selections. From this ideal system we construct a class of admissible routing policies, the overflow routing class, that approximates the ideal situation for each server. Finally, we evaluate the policies and compare them to the lower bound.

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Lee, Duan-Shin. "Analysis of a two-queue model with Bernoulli schedules." Journal of Applied Probability 34, no. 1 (March 1997): 176–91. http://dx.doi.org/10.2307/3215185.

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In this paper we analyze a single server two-queue model with Bernoulli schedules. This discipline is very flexible and contains the exhaustive and 1-limited disciplines as special cases. We formulate the queueing system as a Riemann boundary value problem with shift. The boundary value problem is solved by exploring a Fredholm integral equation around the unit circle. Some numerical examples are presented at the end of the paper.

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Lee, Duan-Shin. "Analysis of a two-queue model with Bernoulli schedules." Journal of Applied Probability 34, no. 01 (March 1997): 176–91. http://dx.doi.org/10.1017/s0021900200100804.

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In this paper we analyze a single server two-queue model with Bernoulli schedules. This discipline is very flexible and contains the exhaustive and 1-limited disciplines as special cases. We formulate the queueing system as a Riemann boundary value problem with shift. The boundary value problem is solved by exploring a Fredholm integral equation around the unit circle. Some numerical examples are presented at the end of the paper.

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39

Salehi-Rad, M. R., K. Mengersen, and G. H. Shahkar. "Reservicing some customers inM/G/1queues under three disciplines." International Journal of Mathematics and Mathematical Sciences 2004, no. 32 (2004): 1715–23. http://dx.doi.org/10.1155/s0161171204210389.

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Consider anM/G/1production line in which a production item is failed with some probability and is then repaired. We consider three repair disciplines depending on whether the failed item is repaired immediately or first stockpiled and repaired after all customers in the main queue are served or the stockpile reaches a specified threshold. For each discipline, we find the probability generating function (p.g.f.) of the steady-state size of the system at the moment of departure of the customer in the main queue, the mean busy period, and the probability of the idle period.

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Bertsimas, Dimitris, and Georgia Mourtzinou. "A unified method to analyze overtake free queueing systems." Advances in Applied Probability 28, no. 02 (June 1996): 588–625. http://dx.doi.org/10.1017/s0001867800048631.

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In this paper we demonstrate that the distributional laws that relate the number of customers in the system (queue), L(Q) and the time a customer spends in the system (queue), S(W) under the first-in-first-out (FIFO) discipline are special cases of the H = λG law and lead to a complete solution for the distributions of L, Q, S, W for queueing systems which satisfy distributional laws for both L and Q (overtake free systems). Moreover, in such systems the derivation of the distributions of L, Q, S, W can be done in a unified way. Consequences of the distributional laws include a generalization of PASTA to queueing systems with arbitrary renewal arrivals under heavy traffic conditions, a generalization of the Pollaczek–Khinchine formula to the G//G/1 queue, an extension of the Fuhrmann and Cooper decomposition for queues with generalized vacations under mixed generalized Erlang renewal arrivals, approximate results for the distributions of L, S in a GI/G/∞ queue, and exact results for the distributions of L, Q, S, W in priority queues with mixed generalized Erlang renewal arrivals.

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Bertsimas, Dimitris, and Georgia Mourtzinou. "A unified method to analyze overtake free queueing systems." Advances in Applied Probability 28, no. 2 (June 1996): 588–625. http://dx.doi.org/10.2307/1428073.

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In this paper we demonstrate that the distributional laws that relate the number of customers in the system (queue), L(Q) and the time a customer spends in the system (queue), S(W) under the first-in-first-out (FIFO) discipline are special cases of the H = λG law and lead to a complete solution for the distributions of L, Q, S, W for queueing systems which satisfy distributional laws for both L and Q (overtake free systems). Moreover, in such systems the derivation of the distributions of L, Q, S, W can be done in a unified way. Consequences of the distributional laws include a generalization of PASTA to queueing systems with arbitrary renewal arrivals under heavy traffic conditions, a generalization of the Pollaczek–Khinchine formula to the G//G/1 queue, an extension of the Fuhrmann and Cooper decomposition for queues with generalized vacations under mixed generalized Erlang renewal arrivals, approximate results for the distributions of L, S in a GI/G/∞ queue, and exact results for the distributions of L, Q, S, W in priority queues with mixed generalized Erlang renewal arrivals.

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42

Harrison, P. G., and E. Pitel. "Sojourn times in single-server queues by negative customers." Journal of Applied Probability 30, no. 4 (December 1993): 943–63. http://dx.doi.org/10.2307/3214524.

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We derive expressions for the Laplace transform of the sojourn time density in a single-server queue with exponential service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. We compare first-come first-served and last-come first-served queueing disciplines for the positive customers, combined with elimination of the last customer in the queue or the customer in service by a negative customer. We also derive the corresponding result for processor-sharing discipline with random elimination. The results show differences not only in the Laplace transforms but also in the means of the distributions, in contrast to the case where there are no negative customers. The various combinations of queueing discipline and elimination strategy are ranked with respect to these mean values.

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43

Harrison, P. G., and E. Pitel. "Sojourn times in single-server queues by negative customers." Journal of Applied Probability 30, no. 04 (December 1993): 943–63. http://dx.doi.org/10.1017/s0021900200044685.

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We derive expressions for the Laplace transform of the sojourn time density in a single-server queue with exponential service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. We compare first-come first-served and last-come first-served queueing disciplines for the positive customers, combined with elimination of the last customer in the queue or the customer in service by a negative customer. We also derive the corresponding result for processor-sharing discipline with random elimination. The results show differences not only in the Laplace transforms but also in the means of the distributions, in contrast to the case where there are no negative customers. The various combinations of queueing discipline and elimination strategy are ranked with respect to these mean values.

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44

Mandjes, M., and M. Nuyens. "SOJOURN TIMES IN THE M/G/1 FB QUEUE WITH LIGHT-TAILED SERVICE TIMES." Probability in the Engineering and Informational Sciences 19, no. 3 (June 22, 2005): 351–61. http://dx.doi.org/10.1017/s0269964805050205.

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The asymptotic decay rate of the sojourn time of a customer in the stationary M/G/1 queue under the foreground–background (FB) service discipline is studied. The FB discipline gives service to those customers that have received the least service so far. We prove that for light-tailed service times, the decay rate of the sojourn time is equal to the decay rate of the busy period. It is shown that FB minimizes the decay rate in the class of work-conserving disciplines.

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45

Koole, Ger, Misja Nuyens, and Rhonda Righter. "The effect of service time variability on maximum queue lengths in MX/G/1 queues." Journal of Applied Probability 42, no. 03 (September 2005): 883–91. http://dx.doi.org/10.1017/s0021900200000875.

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We study the impact of service time distributions on the distribution of the maximum queue length during a busy period for the M X /G/1 queue. The maximum queue length is an important random variable to understand when designing the buffer size for finite-buffer (M/G/1/n) systems. We show the somewhat surprising result that, for three variations of the preemptive last-come–first-served discipline, the maximum queue length during a busy period is smaller when service times are more variable (in the convex sense).

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46

Koole, Ger, Misja Nuyens, and Rhonda Righter. "The effect of service time variability on maximum queue lengths in MX/G/1 queues." Journal of Applied Probability 42, no. 3 (September 2005): 883–91. http://dx.doi.org/10.1239/jap/1127322037.

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We study the impact of service time distributions on the distribution of the maximum queue length during a busy period for the MX/G/1 queue. The maximum queue length is an important random variable to understand when designing the buffer size for finite-buffer (M/G/1/n) systems. We show the somewhat surprising result that, for three variations of the preemptive last-come–first-served discipline, the maximum queue length during a busy period is smaller when service times are more variable (in the convex sense).

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47

Brugno, A., A. N. Dudin, and R. Manzo. "Retrial queue with discipline of adaptive permanent pooling." Applied Mathematical Modelling 50 (October 2017): 1–16. http://dx.doi.org/10.1016/j.apm.2017.05.019.

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48

Li, Jingwen. "From FIFO to LIFO: a functional ordering of service delay via arrival discipline." Journal of Applied Probability 33, no. 2 (September 1996): 507–12. http://dx.doi.org/10.2307/3215074.

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Vasicek (1977) proved that among all queueing disciplines that do not change the departure process of the queue, FIFO and LIFO yield, respectively, the smallest and the largest expectation of any given convex function of the service delay. In this note we further show that, if arriving customers join the queue stochastically ‘closer' to the server(s), then the expected value of any convex function of service delay is larger. As a more interesting result, we also show that if the function under consideration is concave, then the conclusion will be exactly the opposite. This result indicates that LIFO will be the best discipline if the delay cost is an increasing function but at a diminishing rate.

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49

Li, Jingwen. "From FIFO to LIFO: a functional ordering of service delay via arrival discipline." Journal of Applied Probability 33, no. 02 (June 1996): 507–12. http://dx.doi.org/10.1017/s0021900200099927.

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Vasicek (1977) proved that among all queueing disciplines that do not change the departure process of the queue, FIFO and LIFO yield, respectively, the smallest and the largest expectation of any given convex function of the service delay. In this note we further show that, if arriving customers join the queue stochastically ‘closer' to the server(s), then the expected value of any convex function of service delay is larger. As a more interesting result, we also show that if the function under consideration is concave, then the conclusion will be exactly the opposite. This result indicates that LIFO will be the best discipline if the delay cost is an increasing function but at a diminishing rate.

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50

RAZUMCHIK, ROSTISLAV V. "ANALYSIS OF FINITE CAPACITY QUEUE WITH NEGATIVE CUSTOMERS AND BUNKER FOR OUSTED CUSTOMERS USING CHEBYSHEV AND GEGENBAUER POLYNOMIALS." Asia-Pacific Journal of Operational Research 31, no. 04 (August 2014): 1450029. http://dx.doi.org/10.1142/s0217595914500298.

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Consideration is given to the queueing system with incoming Poisson flows of regular and negative customers. Regular customers await service in buffer of finite size r. Each negative customer upon arrival pushes a regular customer out of the queue in buffer (if it is not empty) and moves it to another queue of finite capacity r (bunker). Customers from both queues are served according to exponential distribution with parameter μ, first-come, first-served discipline, but customers in bunker are served with relative priority. Using method based on Chebyshev and Gegenbauer polynomials the algorithm for computation of stationary blocking probabilities and joint probability distribution of the number of customers in buffer and bunker is obtained. Numerical example is provided.

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