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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 (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.
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 (1991): 198–209. http://dx.doi.org/10.1017/s0021900200039528.
Full textAbstract:
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.
3
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 (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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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 (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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Gast, Nicolas, and Benny Van Houdt. "Approximations to Study the Impact of the Service Discipline in Systems with Redundancy." Proceedings of the ACM on Measurement and Analysis of Computing Systems 8, no. 1 (2024): 1–33. http://dx.doi.org/10.1145/3639040.
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As job redundancy has been recognized as an effective means to improve performance of large-scale computer systems, queueing systems with redundancy have been studied by various authors. Existing results include methods to compute the queue length distribution and response time but only when the service discipline is First-Come-First-Served (FCFS). For other service disciplines, such as Processor Sharing (PS), or Last-Come-First-Served (LCFS), only the stability conditions are known. In this paper we develop the first methods to approximate the queue length distribution in a queueing system with redundancy under various service disciplines. We focus on a system with exponential job sizes, i.i.d. copies, and a large number of servers. We first derive a mean field approximation that is independent of the scheduling policy. In order to study the impact of service discipline, we then derive refinements of this approximation to specific scheduling policies. In the case of Processor Sharing, we provide a pair and a triplet approximation. The pair approximation can be regarded as a refinement of the classic mean field approximation and takes the service discipline into account, while the triplet approximation further refines the pair approximation. We also develop a pair approximation for three other service disciplines: First-Come-First-Served, Limited Processor Sharing and Last-Come-First-Served. We present numerical evidence that shows that all the approximations presented in the paper are highly accurate, but that none of them are asymptotically exact (as the number of servers goes to infinity). This makes these approximations suitable to study the impact of the service discipline on the queue length distribution. Our results show that FCFS yields the shortest queue length, and that the differences are more substantial at higher loads.
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Felysia, Novia, Sri Wahyuningsih, and Yuki Novia Nasution. "Analisis Sistem Antrean Untuk Optimalisasi Jumlah Server Menggunakan Model Keputusan Tingkat Aspirasi." EKSPONENSIAL 12, no. 2 (2021): 153. http://dx.doi.org/10.30872/eksponensial.v12i2.808.
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This research aims to obtain information on the number of cashiers with optimal performance in the fast food restaurant at Samarinda Central Plaza queue system using R Studio software. The queue model applied in this research is (M1/M2/c):(FCFS/∞/∞) with First Come First Served (FCFS) queue discipline. The optimal number of cashiers are determinated by using the aspiration level decision model. The results of the analysis showed that the optimal number of cashiers was 3 cashiers with an average number of customers in the queue as many as 1 customer every 15 minutes, the average number of customers in the queue system as many as 2 customers every 15 minutes, the average waiting time for customers in the queue for 1.2 minutes, the average waiting time of customers in the queue system for 6 minutes, and the percentage of idle time is 2.26% that is about 20.34 seconds
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Shalmon, Michael. "Analysis of the GI/GI/1 Queue and its Variations via the LCFS Preemptive Resume Discipline and Its Random Walk Interpretation." Probability in the Engineering and Informational Sciences 2, no. 2 (1988): 215–30. http://dx.doi.org/10.1017/s0269964800000747.
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The unfinished work at arrival instants (FCFS waiting time) can be computed via the LCFS preemptive resume discipline. Analyzing the GI/GI/1 queue in this way gives full interpretation in queuing theoretic language to Feller's analysis of the fluctuations of the random walk. Also, this approach leads naturally to stochastic decompositions for the queue with set-up times, and for other variations of the standard queue. For the M/G/1 queue, the derivations are qualitative, and there are additional connections to branching processes.
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Nurahmadiningsi Nurahmadiningsi, Nadia Manurung, Nico Dhemus, Dinda Setyasari, Dittha Kumari, and Ni Ketut Tari Tastrawati. "Penerapan Model Multi Channel Single Phase dalam Optimisasi Layanan Restoran Cepat Saji." Gemawisata: Jurnal Ilmiah Pariwisata 20, no. 3 (2024): 42–47. http://dx.doi.org/10.56910/gemawisata.v20i3.374.
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A queue is a waiting line from a service facility. Long queues are caused by the number of consumers who come more than the number of service facilities, it is necessary to apply a queuing model to overcome long queues. This research was conducted at Mc Donald's Jimbaran fast food restaurant because the fast food restaurant is located in a tourist and lecture area, causing queues. The current condition at Mcdonald’s Jimbaran there are more than one service facility, the discipline of the first come first service (FCFS) queue is in accordance with the single-phase multi-channel. After analyzed with data from 31 May 2022 to 06 June 2022 for 7 days. Based on the results of the analysis obtained, it can be concluded that at lunch time requires 4 services while at dinner time only requires 2 service.
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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 (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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Saini, Aarti, Deepak Gupta, and A. K. Tripathi. "Analytical Study of Two Serial Channels with Priority and Reneging." International Journal on Recent and Innovation Trends in Computing and Communication 11, no. 6 (2023): 89–93. http://dx.doi.org/10.17762/ijritcc.v11i6.7237.
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Most of the studies of queuing theory, which are useful in our daily life has been investigated by many researchers. The present research is the study of pre-emptive priority queuing system consisting two serial channels in stochastic environment. The impatient behavior of customer’s will be discussed with exponential service distribution and Poisson arrivals. Higher priority customers have pre-emptive priority over the low priority customers. The G.F. technique is used to derive the performance measures of high & low priority queues and assuming FCFS discipline in busy schedule of higher priority class. Also evaluate queue behavior graphically and discussed a special case at the end which shows utilization of channels.
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Li, Tao, Liyuan Zhang, and Shan Gao. "An M/G/1 retrial queue with single working vacation under Bernoulli schedule." RAIRO - Operations Research 54, no. 2 (2020): 471–88. http://dx.doi.org/10.1051/ro/2019008.
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In this paper, an M/G/1 retrial queue with general retrial times and single working vacation is considered. We assume that the customers who find the server busy are queued in the orbit in accordance with a first-come-first-served (FCFS) discipline and only the customer at the head of the queue is allowed access to the server. During the normal period, if the orbit queue is not empty at a service completion instant, the server begins a working vacation with specified probability q (0 ≤ q ≤ 1), and with probability 1 − q, he waits for serving the next customer. During the working vacation period, customers can be served at a lower service rate. We first present the necessary and sufficient condition for the system to be stable. Using the supplementary variable method, we deal with the generating functions of the server state and the number of customers in the orbit. Various interesting performance measures are also derived. Finally, some numerical examples and cost optimization analysis are presented.
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Kulkarni, V. G., and K. D. Glazebrook. "Output analysis of a single-buffer multiclass queue: FCFS service." Journal of Applied Probability 39, no. 2 (2002): 341–58. http://dx.doi.org/10.1239/jap/1025131430.
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We consider an infinite capacity buffer where the incoming fluid traffic belongs to K different types modulated by K independent Markovian on-off processes. The kth input process is described by three parameters: (λk, μk, rk), where 1/λk is the mean off time, 1/μk is the mean on time, and rk is the constant peak rate during the on time. The buffer empties the fluid at rate c according to a first come first served (FCFS) discipline. The output process of type k fluid is neither Markovian, nor on-off. We approximate it by an on-off process by defining the process to be off if no fluid of type k is leaving the buffer, and on otherwise. We compute the mean on time τkon and mean off time τkoff. We approximate the peak output rate by a constant rko so as to conserve the fluid. We approximate the output process by the three parameters (λko, μko, rko), where λko = 1/τkoff, and μko = 1/τkon. In this paper we derive methods of computing τkon, τkoff and rko for k = 1, 2,…, K. They are based on the results for the computation of expected reward in a fluid queueing system during a first passage time. We illustrate the methodology by a numerical example. In the last section we conduct a similar output analysis for a standard M/G/1 queue with K types of customers arriving according to independent Poisson processes and requiring independent generally distributed service times, and following a FCFS service discipline. For this case we are able to get analytical results.
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Kulkarni, V. G., and K. D. Glazebrook. "Output analysis of a single-buffer multiclass queue: FCFS service." Journal of Applied Probability 39, no. 02 (2002): 341–58. http://dx.doi.org/10.1017/s0021900200022555.
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We consider an infinite capacity buffer where the incoming fluid traffic belongs to K different types modulated by K independent Markovian on-off processes. The kth input process is described by three parameters: (λ k , μ k , r k ), where 1/λ k is the mean off time, 1/μ k is the mean on time, and r k is the constant peak rate during the on time. The buffer empties the fluid at rate c according to a first come first served (FCFS) discipline. The output process of type k fluid is neither Markovian, nor on-off. We approximate it by an on-off process by defining the process to be off if no fluid of type k is leaving the buffer, and on otherwise. We compute the mean on time τ k on and mean off time τ k off. We approximate the peak output rate by a constant r k o so as to conserve the fluid. We approximate the output process by the three parameters (λ k o, μ k o, r k o), where λ k o = 1/τ k off, and μ k o = 1/τ k on. In this paper we derive methods of computing τ k on, τ k off and r k o for k = 1, 2,…, K. They are based on the results for the computation of expected reward in a fluid queueing system during a first passage time. We illustrate the methodology by a numerical example. In the last section we conduct a similar output analysis for a standard M/G/1 queue with K types of customers arriving according to independent Poisson processes and requiring independent generally distributed service times, and following a FCFS service discipline. For this case we are able to get analytical results.
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Ibrahim, Novita, Salmun K. Nasib, Agusyarif Rezka Nuha, Muh Rifai Katili, Nurwan Nurwan, and Djihad Wungguli. "Analisis Sistem Antrian dengan Model M/M/C dalam Meningkatkan Efektivitas Kinerja Sistem." Algoritma : Jurnal Matematika, Ilmu pengetahuan Alam, Kebumian dan Angkasa 3, no. 2 (2025): 20–34. https://doi.org/10.62383/algoritma.v3i2.431.
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Study This aim For analyze system queue at the Population and Registration Service Civil ( Disdukcapil) Bone Bolango Regency as well as apply the M/M/C (Multi Channel Single Phase ) queuing model optimizing performance system and upgrade effectiveness service to public . Arrival data visitors and time service collected for 5 days through observation . Analysis results show system applied queue moment This is the model (M/M/4 ): (FIFO/∞/∞) with level arrival visitors Poisson distribution , time service distribute Exponential , 4 counters service , discipline first-come first-served (FCFS) queues , as well source arrival and capacity queue No limited . Size performance the system in existing conditions shows time wait for the average visitor in system amounting to 26.4 minutes and time Wait in queue amounting to 44.4 minutes . For optimizing performance , research recommend application of the model (M/M/7 ) : (FIFO/∞/∞) with add amount counter service into 7 counters . In this model , level utility system (ρ) is below 50 % ie about 41%, which is considered effective Because enter in range level utility low (5%-10%). Application of the queuing model with 7 counters projected can shorten time wait for the average visitor in system to 18.42 minutes and time Wait in queue to 18 minutes . Findings This expected can increase effectiveness service and satisfaction public to service Disdukcapil Bone Bolango Regency.
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Salimnejad, Mehrdad, and Nikolaos Pappas. "On the Age of Information in a Two-User Multiple Access Setup." Entropy 24, no. 4 (2022): 542. http://dx.doi.org/10.3390/e24040542.
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This work considers a two-user multiple access channel in which both users have Age of Information (AoI)-oriented traffic with different characteristics. More specifically, the first user has external traffic and cannot control the generation of status updates, and the second user monitors a sensor and transmits status updates to the receiver according to a generate-at-will policy. The receiver is equipped with multiple antennas and the transmitters have single antennas; the channels are subject to Rayleigh fading and path loss. We analyze the average AoI of the first user for a discrete-time first-come-first-served (FCFS) queue, last-come-first-served (LCFS) queue, and queue with packet replacement. We derive the AoI distribution and the average AoI of the second user for a threshold policy. Then, we formulate an optimization problem to minimize the average AoI of the first user for the FCFS and LCFS with preemption queue discipline to maintain the average AoI of the second user below a given level. The constraints of the optimization problem are shown to be convex. It is also shown that the objective function of the problem for the first-come-first-served queue policy is non-convex, and a suboptimal technique is introduced to effectively solve the problem using the algorithms developed for solving a convex optimization problem. Numerical results illustrate the performance of the considered optimization algorithm versus the different parameters of the system. Finally, we discuss how the analytical results of this work can be extended to capture larger setups with more than two users.
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Ramasamy, Sivasamy, Onkabetse A. Daman, and Sulaiman Sani. "An M/G/2 queue where customers are served subject to a minimum violation of FCFS queue discipline." European Journal of Operational Research 240, no. 1 (2015): 140–46. http://dx.doi.org/10.1016/j.ejor.2014.06.048.
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Bataona, Benediktus L. V., Antonio E. L. Nyoko, and Ni Putu Nursiani. "ANALISIS SISTEM ANTRIAN DALAM OPTIMALISASI LAYANAN DI SUPERMARKET HYPERSTORE." Journal of Management : Small and Medium Enterprises (SMEs) 12, no. 2 (2020): 225–37. http://dx.doi.org/10.35508/jom.v12i2.2695.
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The purpose of this study is to determine the average customer arrival level and average service level at the Hyperstore Supermarket and analyze the queuing system by optimizing the number of cashiers that must be set at the Hyperstore Supermarket. The results showed that the queuing model used at the Hyperstore Supermarket is Multi Channel Single Phase by applying the First Come First Serve (FCFS) queuing discipline. The average service level at the Hyperstore Supermarket is 146 people/hour. The current number of lines opened at the Hyperstore Supermarket is 6 cashiers. However, 6 cashier lines are excess even though 3 cashier lines are sufficient. Queue system analysis shows that the optimal number of cashier lines which is 1 cashier line open at 3:00 pm to 4:00, 2 cashier lines open at 16:00 to 19:00 and 3 cashiers lines opened at 19:00 to 20:00
 Keywords: Queue Theory, Multiple Line Queue Model, Cashier, Service Optimization
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Ramadhana, Nur Asmi. "ANALISIS TINGKAT UTILITAS SISTEM ANTRIAN PADA PT POS INDONESIA KOTA PAREPARE." DECISION : Jurnal Ekonomi dan Bisnis 2, no. 1 (2021): 53–61. http://dx.doi.org/10.31850/decision.v2i1.736.
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This study aims to analyze the utility level of the service queue system at PT. Pos Indonesia Kota Parepare. This research supports the analysis of the utility level of the service queue system at PT. Pos Indonesia, the city of Parepare. From the analysis results obtained queuing system at PT. Pos Indonesia, Kota Parepare M / M / S double-track queue model. The results of calculations with the M / M / S queuing model at PT. Pos Indonesia, Kota Parepare implements queuing discipline, namely First Come First Server (FCFS). Patterns of the arrival of customers, distribution of passion from the results of calculations with the dual-line queuing system model (M / M / S), it is known that PO is 3.90%, while the post for 50% of the time is free of 50% (free service hours). Ls 5,423 people or five people. Lq 3,423 people or three people, Ws 0.112 hours or 6.7 minutes, while Wq 0.071 hours or 4.27 minutes.
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Giorno, V., A. G. Nobile, and L. M. Ricciardi. "On some time-non-homogeneous diffusion approximations to queueing systems." Advances in Applied Probability 19, no. 4 (1987): 974–94. http://dx.doi.org/10.2307/1427111.
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Time-non-homogeneous diffusion approximations to single server–single queue–FCFS discipline systems are considered. Under various assumptions on the nature of the time-dependent functions appearing in the infinitesimal moments the transient and the regime behaviour of the approximating diffusions are analysed in some detail. Special attention is then given to the study of a diffusion approximation characterized by a linear drift and by a periodically time-varying infinitesimal variance. Unlike the behaviour of transition functions and moments, the p.d.f. of the busy period is seen to be unaffected by the presence of such periodicity.
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Giorno, V., A. G. Nobile, and L. M. Ricciardi. "On some time-non-homogeneous diffusion approximations to queueing systems." Advances in Applied Probability 19, no. 04 (1987): 974–94. http://dx.doi.org/10.1017/s0001867800017523.
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Time-non-homogeneous diffusion approximations to single server–single queue–FCFS discipline systems are considered. Under various assumptions on the nature of the time-dependent functions appearing in the infinitesimal moments the transient and the regime behaviour of the approximating diffusions are analysed in some detail. Special attention is then given to the study of a diffusion approximation characterized by a linear drift and by a periodically time-varying infinitesimal variance. Unlike the behaviour of transition functions and moments, the p.d.f. of the busy period is seen to be unaffected by the presence of such periodicity.
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Sağlam, Vedat, and Müjgan Zobu. "A Two-Stage Model Queueing with No Waiting Line between Channels." Mathematical Problems in Engineering 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/679369.
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We consider a new queuing model with sequential two stations (stages), single server at each station, where no queue is allowed at station 2 and with no restriction at station 1. There is a FCFS service discipline in which the input stream is Poisson having rate . The service time of any customer at server () is exponential with parameter . The state probabilities and loss probability of this model are given. The performance measures are obtained and optimized, and, additionally, the model is simulated. The simulation results, exact results, and optimal results of the performance measures are numerically computed for different parameters.
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Juneja, Sandeep, and Tushar Raheja. "The Concert Queueing Game: Fluid Regime with Random Order Service." International Game Theory Review 17, no. 02 (2015): 1540012. http://dx.doi.org/10.1142/s0219198915400125.
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The concert queueing problem corresponds to determining the equilibrium arrival profile of non-cooperative customers selecting their arrival times to a queue where the service opens at a specified time. The customers are allowed to arrive before or after this time. This problem has a variety of queuing applications including how people queue at airport, movie theaters, passport offices, ration lines, etc. This also captures the settings where large computational jobs are sent to servers that open for service at a specified time. Substantial literature is devoted to studying the more tractable fluid version of this problem, that is, each customer is considered an infinitesimal particle, resulting in a non-atomic game between customers. This allows for explicit determination of the unique equilibrium arrival profile in many such settings as well as the associated socially optimal centralized solution. The knowledge of both then allows the computation of price of anarchy (PoA) in the system. The literature thus far focuses on queues with the first come first serve (FCFS) service discipline. In this paper, we again consider the fluid regime and extend the analysis to the case where the service discipline is random order service (ROS). This is equivalent to the practically equally important processor sharing regime when the service times are exponential. The latter is relevant in computational settings while the former is a good approximation to settings where a customer is selected more or less at random by the server.
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S, Nivetha Therasal, and Thiagarajan M. "Poisson Input and Exponential Service Time Finite Capacity Interdependent Queueing Model with Breakdown and Controllable Arrival Rates." Indian Journal of Science and Technology 17, no. 12 (2024): 1167–79. https://doi.org/10.17485/IJST/v17i12.2852.
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Abstract <strong>Objectives:</strong> This study aims at (i) introducing the finite capacity of the interdependent queueing model with breakdown and controllable arrival rates, (ii) calculating the average number of clients in the system, and identifying the expected waiting period of the clients in the system, (iii) dealing with the model descriptions, steady-state equations, and characteristics, which are expressed in terms of , and (iv) analyzing the probabilities of the queueing system and its characteristics with numerical verification of the obtained results. <strong>Methods:</strong> While providing the input, the arrival rates through faster and slower arrival rates are controlled using the Poisson process. Also, the service provides an exponential distribution. The server provides the service on an FCFS basis. In this article, two types of models are used: and which are the system’s conditions, where represents the number of units present in the queue in which their probability is and . All probabilities are distributed based on the speed of advent using this concept. Then, the steady-state probabilities are computed using a recursive approach. <strong>Findings:</strong> This paper discovers the number of clients using the system on average and the expected number of clients in the system using the probability of the steady-state calculation. Little’s formula is used to derive the expected waiting period of the clients in the system.<strong> Novelty:</strong> There are articles connected to the finite capacity of failed service in functioning and malfunctioning, but this takes the initiative to provide a link in connection with the rates of the controllable arrivals and interdependency in the arrival and service processes. Mathematics Subject allocation: 60K25, 68M20, 90B22. <strong>Keywords:</strong> M/M/1/K Queue Model, Finite Capacity, Breakdown, Controllable Arrival rates, FCFS Queue Discipline
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Brooms, A. C. "On the Nash equilibria for the FCFS queueing system with load-increasing service rate." Advances in Applied Probability 37, no. 2 (2005): 461–81. http://dx.doi.org/10.1239/aap/1118858634.
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We consider a service system (QS) that operates according to the first-come-first-served (FCFS) discipline, and in which the service rate is an increasing function of the queue length. Customers arrive sequentially at the system, and decide whether or not to join using decision rules based upon the queue length on arrival. Each customer is interested in selecting a rule that meets a certain optimality criterion with regard to their expected sojourn time in the system; as a consequence, the decision rules of other customers must be taken into account. Within a particular class of decision rules for an associated infinite-player game, the structure of the Nash equilibrium routeing policies is characterized. We prove that, within this class, there exist a finite number of Nash equilibria, and that at least one of these is nonrandomized. Finally, with the aid of simulation experiments, we explore the extent to which the Nash equilibria are characteristic of customer joining behaviour under a learning rule based on system-wide data.
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Brooms, A. C. "On the Nash equilibria for the FCFS queueing system with load-increasing service rate." Advances in Applied Probability 37, no. 02 (2005): 461–81. http://dx.doi.org/10.1017/s0001867800000264.
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We consider a service system (QS) that operates according to the first-come-first-served (FCFS) discipline, and in which the service rate is an increasing function of the queue length. Customers arrive sequentially at the system, and decide whether or not to join using decision rules based upon the queue length on arrival. Each customer is interested in selecting a rule that meets a certain optimality criterion with regard to their expected sojourn time in the system; as a consequence, the decision rules of other customers must be taken into account. Within a particular class of decision rules for an associated infinite-player game, the structure of the Nash equilibrium routeing policies is characterized. We prove that, within this class, there exist a finite number of Nash equilibria, and that at least one of these is nonrandomized. Finally, with the aid of simulation experiments, we explore the extent to which the Nash equilibria are characteristic of customer joining behaviour under a learning rule based on system-wide data.
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Sirait, Paskah Riny, and Parapat Gultom. "Analisis Sistem Antrian pada Bank Negara Indonesia Kantor Cabang Kawasan Industri Medan." EduMatSains : Jurnal Pendidikan, Matematika dan Sains 7, no. 2 (2023): 292–304. http://dx.doi.org/10.33541/edumatsains.v7i2.4283.
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The problem that occurs at Bank Negara Indonesia of Medan Industrial Area Branch Office is the occurrence of unemployed tellers at certain hours. The purpose of this study is to determine the queue system at Bank Negara Indonesia of Medan Industrial Area Branch Office and to find out the optimal number of tellers at the bank. The structure of the queue model that occurs at Bank Negara Indonesia of The Medan Industrial Area Branch Office is a Multi Channel-Single Phase by applying the discipline of the First Come First Serve queue. The bank has 4 tellers with an average customer arrival value (λ) of 20 people per hour and an average service level (μ) of 22 people per hour. The results of data processing show the arrival pattern of Poisson-distributed customers and the general distribution service pattern. This study resulted in a queueing model (M / G / 4): (FCFS / ∞ / ∞). The optimal number of tellers in providing customer service is to reduce 1 teller. Based on the calculation results with only 3 tellers, it can reduce the probability of the teller being unemployed from the initial one by 40.25% to 39.97%.
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Therasal, S. Nivetha, and M. Thiagarajan. "Poisson Input and Exponential Service Time Finite Capacity Interdependent Queueing Model with Breakdown and Controllable Arrival Rates." Indian Journal Of Science And Technology 17, no. 12 (2024): 1167–79. http://dx.doi.org/10.17485/ijst/v17i12.2852.
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Objectives: This study aims at (i) introducing the finite capacity of the interdependent queueing model with breakdown and controllable arrival rates, (ii) calculating the average number of clients in the system, and identifying the expected waiting period of the clients in the system, (iii) dealing with the model descriptions, steady-state equations, and characteristics, which are expressed in terms of , and (iv) analyzing the probabilities of the queueing system and its characteristics with numerical verification of the obtained results. Methods: While providing the input, the arrival rates through faster and slower arrival rates are controlled using the Poisson process. Also, the service provides an exponential distribution. The server provides the service on an FCFS basis. In this article, two types of models are used: and which are the system’s conditions, where represents the number of units present in the queue in which their probability is and . All probabilities are distributed based on the speed of advent using this concept. Then, the steady-state probabilities are computed using a recursive approach. Findings: This paper discovers the number of clients using the system on average and the expected number of clients in the system using the probability of the steady-state calculation. Little’s formula is used to derive the expected waiting period of the clients in the system. Novelty: There are articles connected to the finite capacity of failed service in functioning and malfunctioning, but this takes the initiative to provide a link in connection with the rates of the controllable arrivals and interdependency in the arrival and service processes. Mathematics Subject allocation: 60K25, 68M20, 90B22. Keywords: M/M/1/K Queue Model, Finite Capacity, Breakdown, Controllable Arrival rates, FCFS Queue Discipline
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Mélange, Willem, Joris Walraevens, Dieter Claeys, Bart Steyaert, and Herwig Bruneel. "The impact of a global FCFS service discipline in a two-class queue with dedicated servers." Computers & Operations Research 71 (July 2016): 23–33. http://dx.doi.org/10.1016/j.cor.2016.01.005.
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Fatimah, Fatimah, Arfianty Arfianty, and Nelly Agustina. "UTILITAS SISTEM ANTRIAN PADA PT. PEGADAIAN (PERSERO) CABANG RAPPANG." Economos : Jurnal Ekonomi dan Bisnis 4, no. 2 (2021): 89–97. http://dx.doi.org/10.31850/economos.v4i2.819.
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This study aims to determine the level of utility queuing system at PT. Pegadaian (Persero) Rappang Branch. The data collection techniques used in this study are interviews and observation. The population used in this study were all customers who came to PT. Pegadaian (Persero) Rappang Branch at 08.00-12.00 and 13.00-15.00. The data analysis technique in this study uses a descriptive method, which is a way of analyzing data by comparing the problem with existing theories, besides that, a formula is also used to solve the queue problem in PT. Pegadaian (Persero) Rappang Branch. The results of calculations using the M/M/S queuing method at PT. Pegadaian (Persero) Rappang Branch applies queue discipline, namely First Come First Server (FCFS). From the results of calculations using the queuing system method with multiple lines (M/M/S), it is known that the probability that there are 0 people in the system (no customers in the system) P0 is 8.78%, while the utility level of pawnshops/busyness P is 68% of the time that does not work. used 32% (empty service hours).
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Hasan, Raed Abdulkareem, Omar Ibrahim Obaid, Ans Ibrahim Mahameed AlQassab, Husniyah Jasim, and Saad Ahmed Dheyab. "Mapping Web Service Characteristics to Queueing Theory Models for Performance Analysis." Babylonian Journal of Networking 2024 (July 20, 2024): 98–110. http://dx.doi.org/10.58496/bjn/2024/011.
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The article presents a comprehensive analysis of mapping the characteristics of Web services to queueing theory models, specifically the M/M/1 queueing model. The authors aim to establish a theoretical foundation for modeling the performance of Web services and deriving response time formulae. The key points covered in the article are as follows: 1. Analyzing the characteristics of Web services: The article examines the six fundamental characteristics of queueing systems, including arrival patterns, service patterns, queue discipline, system capacity, and the number of servers, and maps them to the Web service environment. 2. Representation using Kendall's notation: The Web service characteristics are represented using Kendall's notation, resulting in the M/M/1/∞/FCFS model, where M denotes exponential distributions for arrival and service patterns, 1 represents a single server, ∞ represents infinite system capacity, and FCFS represents the first-come, first-served queue discipline. 3. Derivation of response time formulae: The article provides a detailed derivation of the response time formulae for the M/M/1 model using stochastic processes, Markov chains, and the birth-death process. The derivation incorporates steady-state conditions and results in a formula that calculates the probability of completing a request within a user-specified response time. 4. Practical applications: The derived response time formula can be used by service providers to determine the probability of completing user requests within specified Service Level Agreements (SLAs). By setting a threshold value (filter value), the service provider can select requests with a higher probability of completion for further processing. Overall, the article contributes to the understanding of Web service performance modeling by establishing a theoretical foundation based on queueing theory. The mapping of Web service characteristics to the M/M/1 model and the derivation of response time formulae provide valuable insights for service providers to analyze and optimize the performance of their Web services while adhering to SLAs
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Rambe, Aida Zahrah Hasanati Br, Fatimah Zahra Rambe, Jihan Azzahra, Lidia Sartika Gultom, and Putri Chairunnisa. "Analisis Model Antrian Multi Channel-Single Phase Pada Doorsmeer Mobil dengan Menggunakan Software POM-QM for Windows." Indo-MathEdu Intellectuals Journal 5, no. 1 (2024): 602–11. http://dx.doi.org/10.54373/imeij.v5i1.780.
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Queuing is the activity of a group of people or goods in a waiting line to wait for a service. The queueing system can be interpreted as the arrival of customers for a service, waiting to get service, and leaving the system after getting service. This research is applied research using the observation method. This research aims to provide solutions to queuing problems at Agung Auto Service car doorsmer Jl. S.M. Raja No. 3, Rantauprapat, North Sumatra. Data is taken by paying attention to the time of arrival of customers and how many customers can be served. This car doorman applies the First Come, First Served (FCFS) discipline, where the first customer to arrive will get the first service through the queuing process. Therefore, the model that can be used is the multi-channel single-phase model, where there are two or more service facilities flowed by a single queue. From the results of the calculation of the model simulation in POM-QM for Windows, it can be seen that the performance of the service system at the Agung Auto Service car doorsmeer is not optimal because initially there were only 3 service servers, causing queues. Therefore, it is necessary to add up to six service servers to produce optimal service
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Li, Jianjun, and Liwei Liu. "On an M/G/1 queue in random environment with Min(N, V) policy." RAIRO - Operations Research 52, no. 1 (2018): 61–77. http://dx.doi.org/10.1051/ro/2018006.
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In this paper, we analyze an M∕G∕1 queue operating in multi-phase random environment with Min(N, V) vacation policy. In operative phase i, i = 1, 2, …, n, customers are served according to the discipline of First Come First Served (FCFS). When the system becomes empty, the server takes a vacation under the Min(N, V) policy, causing the system to move to vacation phase 0. At the end of a vacation, if the server finds no customer waiting, another vacation begins. Otherwise, the system jumps from the phase 0 to some operative phase i with probability qi, i = 1, 2, …, n. And whenever the number of the waiting customers in the system reaches N, the server interrupts its vacation immediately and the system jumps from the phase 0 to some operative phase i with probability qi, i = 1, 2, …, n, too. Using the method of supplementary variable, we derive the distribution for the stationary system size at arbitrary epoch. We also obtain mean system size, the results of the cycle analysis and the sojourn time distribution. In addition, some special cases and numerical examples are presented.
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Seshadri, Sridhar. "A sample path analysis of the delay in the M/G/C system." Journal of Applied Probability 33, no. 1 (1996): 256–66. http://dx.doi.org/10.2307/3215282.
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Using sample path analysis we show that under the same load the mean delay in queue in the M/G/2 system is smaller than that in the corresponding M/G/1 system, when the service time has either the DMRL or NBU property and the service discipline is FCFS. The proof technique uses a new device that equalizes the work in a two server system with that in a single sterver system. Other interesting quantities such as the average difference in work between the two servers in the GI/G/2 system and an exact alternate derivation of the mean delay in the M/M/2 system from sample path analysis are presented. For the same load, we also show that the mean delay in the M/G/C system with general service time distribution is smaller than that in the M/G/1 system when the traffic intensity is less than 1/c.
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Seshadri, Sridhar. "A sample path analysis of the delay in the M/G/C system." Journal of Applied Probability 33, no. 01 (1996): 256–66. http://dx.doi.org/10.1017/s0021900200103900.
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Using sample path analysis we show that under the same load the mean delay in queue in the M/G/2 system is smaller than that in the corresponding M/G/1 system, when the service time has either the DMRL or NBU property and the service discipline is FCFS. The proof technique uses a new device that equalizes the work in a two server system with that in a single sterver system. Other interesting quantities such as the average difference in work between the two servers in the GI/G/2 system and an exact alternate derivation of the mean delay in the M/M/2 system from sample path analysis are presented. For the same load, we also show that the mean delay in the M/G/C system with general service time distribution is smaller than that in the M/G/1 system when the traffic intensity is less than 1/c.
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Naresh, Kumar Naresh Kumar. "Machine Repair Problem with Additional Repairman and Mixed Standbys." Online International Interdisciplinary Research Journal 08 (April 5, 2018): 06–11. https://doi.org/10.5281/zenodo.14875991.
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This paper is concerned with the analysis of a multi-component state dependent machine repair problem consisting of M operating units with two types of spares in the system to ensure the desired reliability for system. There is a provision of repair facility containing permanent as well as the additional repairman in order to provide repair to failed units one by one according to first come first served (FCFS) discipline. When all the spare units are exhausted then the system works in short mode; in this case a failed unit works with degraded rate due to stress. The life-time and repair time are both exponentially distributed. The recursive technique is used to find out the steady-state probabilities. Some system indices such as expected number of failed units in thesystem, expected number of cold as well as warm standby units, etc. are derived in terms of probabilities.
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Atencia-Mckillop, Iván, Sixto Sánchez-Merino, Inmaculada Fortes-Ruiz, and José Luis Galán-García. "Discrete-Time Retrial Queuing Systems with Last-Come-First-Served (LCFS) and First-Come-First-Served (FCFS) Disciplines: Negative Customer Impact and Stochastic Analysis." Mathematics 13, no. 1 (2024): 107. https://doi.org/10.3390/math13010107.
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This paper examines a discrete-time retrial queuing system that incorporates negative customers, system breakdowns, and repairs. In this model, an arriving customer has the option to go directly to the server, pushing the currently served customer, if any, to the front of the orbit queue, or to join the orbit based on a First-Come-First-Served (FCFS) discipline. The study also considers negative customers who not only remove the customer currently being served but also cause a server breakdown. An in-depth analysis of the model is conducted using a generating function approach, leading to the determination of the distribution and expected values of the number of customers in the orbit and the entire system. The paper explores the stochastic decomposition law and provides bounds for the difference between the steady-state distribution of this system and a comparable standard system. Recursive formulas for the steady-state distributions of the orbit and the system are developed. Additionally, it is shown that the studied discrete-time system can approximate the M/G/1 continuous-time version of the model. The research includes a detailed examination of the customer’s sojourn time distribution in the orbit and the system, utilizing the busy period of an auxiliary system. The paper concludes with numerical examples that highlight how different system parameters affect various performance characteristics, and a section summarizing the key research contributions.
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ADAN, I. J. B. F., A. SLEPTCHENKO, and G. J. VAN HOUTUM. "REDUCING COSTS OF SPARE PARTS SUPPLY SYSTEMS VIA STATIC PRIORITIES." Asia-Pacific Journal of Operational Research 26, no. 04 (2009): 559–85. http://dx.doi.org/10.1142/s0217595909002377.
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We study static repair priorities in a system consisting of one repair shop and one stockpoint, where spare parts of multiple, critical repairables are kept on stock to serve an installed base of technical systems. Demands for ready-for-use parts occur according to Poisson processes, and are accompanied by returns of failed parts. The demands are met from stock if possible, and otherwise they are backordered and fulfilled as soon as a ready-for-use part becomes available. Returned failed parts are immediately sent into repair. The repairables are assigned to static priority classes. The repair shop is modeled as a single-server queue, where the failed parts are served according to these priority classes. We show that under a given assignment of repairables to priority classes, optimal spare parts stock levels follow from Newsvendor equations. Next, we develop fast and effective heuristics for the assignment of repairables to priority classes. Subsequently, we compare the performance of the system under these static priorities to the case with a First-Come First-Served (FCFS) service discipline. We show that in many cases static priorities reduce total inventory holding and backordering costs by more than 40%. Finally, we analyse the effect of the number of priority classes. We show that 2 priority classes suffice to obtain 90% of the maximal savings via static priorities.
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Ahuja, Anjali, and Anamika Jain. "JOINING STRATEGIES IN MULTIPLE VACATION MACHINING SYSTEM WITH HETEROGENEOUS SERVERS." COMPUSOFT: An International Journal of Advanced Computer Technology 08, no. 07 (2019): 3269–73. https://doi.org/10.5281/zenodo.14832629.
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Markov machine repair model consisting of mixed spares under the supervision of two heterogeneous repairmen is investigated. Every time any component fails, it is quickly supplanted by a spare component if accessible. In the event when all spares are used up and the system operates with less than K functioning components then the failure of components happens in a degraded fashion. The distribution of failure time and repair time of the components are assumed to be exponential and FCFS discipline is used for repairing of failed components. The failed components may balk with constant probability as a result of impatience and may renege according to negative exponential distribution if the repairmen are busy. A reneged component can be retained in the queue by using some convincing ways to finish the repair process. After the repair, each component may rejoin the system as a feedback customer with some probability in case of imperfect repair. The repairmen switch to vacation state from busy state whenever there are no failed components in the system and repairmen switch back to the busy state as soon as any failed component arrives. The numerical technique named ‘Successive Over Relaxation (SOR) Technique’ is employed to acquire the system state probabilities at steady state which are then used to calculate the mean count of failed components in the system, throughput, carried load and other indices of the system. The numerical outcomes acquired are verified using the results generated by ANFIS.
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NI WAYAN EKANTARI, NI WAYAN, NI KETUT TARI TASTRAWATI, and KARTIKA SARI. "PENERAPAN MODEL ANTREAN MULTI CHANNEL SINGLE PHASE PADA SISTEM PELAYANAN RESTORAN CEPAT SAJI." E-Jurnal Matematika 10, no. 3 (2021): 163. http://dx.doi.org/10.24843/mtk.2021.v10.i03.p337.
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A queue will occur if the average number of arrivals exceeds the capacity of service facilities. Fast food restaurants are one of the places that usually have long queues at lunchtime and dinner time. KFC in Bali, located in the village of Sanur, is a fast food restaurant that is experiencing long queues. This is because this restaurant is located in a tourism area and the only KFC outlet on the Ngurah Rai Sanur bypass line and does not yet have a drive-thru service. The current condition at KFC Sanur is that there is more than one service facility, disciplined first come first service (FCFS) queues according to the multi channel single phase queuing model. After being analyzed with data taken before the pandemic period on November 18, 2019 to December 1, 2019 for 14 days during weekdays and weekends, it was found that the performance of the KFC Sanur queue system would have a smaller utility level if there were 3 active server. The total cost per customer if there are 2 active server is IDR 78,692.38 and if there are 3 server is IDR 75,788.45. Based on the results of this analysis, it can be concluded that it will be more optimal if there are 3 active server.
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Hyytiä, Esa, Samuli Aalto, Aleksi Penttinen, and Jorma Virtamo. "On the Value Function of the M/G/1 FCFS and LCFS Queues." Journal of Applied Probability 49, no. 4 (2012): 1052–71. http://dx.doi.org/10.1239/jap/1354716657.
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We consider a single-server queue with Poisson input operating under first-come–first-served (FCFS) or last-come–first-served (LCFS) disciplines. The service times of the customers are independent and obey a general distribution. The system is subject to costs for holding a customer per unit of time, which can be customer specific or customer class specific. We give general expressions for the corresponding value functions, which have elementary compact forms, similar to the Pollaczek–Khinchine mean value formulae. The results generalize earlier work where similar expressions have been obtained for specific service time distributions. The obtained value functions can be readily applied to develop nearly optimal dispatching policies for a broad range of systems with parallel queues, including multiclass scenarios and the cases where service time estimates are available.
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Hyytiä, Esa, Samuli Aalto, Aleksi Penttinen, and Jorma Virtamo. "On the Value Function of the M/G/1 FCFS and LCFS Queues." Journal of Applied Probability 49, no. 04 (2012): 1052–71. http://dx.doi.org/10.1017/s0021900200012870.
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We consider a single-server queue with Poisson input operating under first-come–first-served (FCFS) or last-come–first-served (LCFS) disciplines. The service times of the customers are independent and obey a general distribution. The system is subject to costs for holding a customer per unit of time, which can be customer specific or customer class specific. We give general expressions for the corresponding value functions, which have elementary compact forms, similar to the Pollaczek–Khinchine mean value formulae. The results generalize earlier work where similar expressions have been obtained for specific service time distributions. The obtained value functions can be readily applied to develop nearly optimal dispatching policies for a broad range of systems with parallel queues, including multiclass scenarios and the cases where service time estimates are available.
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Righter, Rhonda, J. George Shanthikumar, and Genji Yamazaki. "On extremal service disciplines in single-stage queueing systems." Journal of Applied Probability 27, no. 2 (1990): 409–16. http://dx.doi.org/10.2307/3214660.
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It is shown that among all work-conserving service disciplines that are independent of the future history, the first-come-first-served (FCFS) service discipline minimizes [maximizes] the average sojourn time in a G/GI/1 queueing system with new better [worse] than used in expectation (NBUE[NWUE]) service time distribution. We prove this result using a new basic identity of G/GI/1 queues that may be of independent interest. Using a relationship between the workload and the number of customers in the system with different lengths of attained service it is shown that the average sojourn time is minimized [maximized] by the least-attained-service time (LAST) service discipline when the service time has the decreasing [increasing] mean residual life (DMRL[IMRL]) property.
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Righter, Rhonda, J. George Shanthikumar, and Genji Yamazaki. "On extremal service disciplines in single-stage queueing systems." Journal of Applied Probability 27, no. 02 (1990): 409–16. http://dx.doi.org/10.1017/s0021900200038869.
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It is shown that among all work-conserving service disciplines that are independent of the future history, the first-come-first-served (FCFS) service discipline minimizes [maximizes] the average sojourn time in a G/GI/1 queueing system with new better [worse] than used in expectation (NBUE[NWUE]) service time distribution. We prove this result using a new basic identity of G/GI/1 queues that may be of independent interest. Using a relationship between the workload and the number of customers in the system with different lengths of attained service it is shown that the average sojourn time is minimized [maximized] by the least-attained-service time (LAST) service discipline when the service time has the decreasing [increasing] mean residual life (DMRL[IMRL]) property.
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Shanthikumar, J. George, and Ushio Sumita. "On the busy-period distributions of M/G/1/K queues by state-dependent arrivals and FCFS/LCFS-P service disciplines." Journal of Applied Probability 22, no. 4 (1985): 912–19. http://dx.doi.org/10.2307/3213958.
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The busy-period distributions of M/G/1/K queues with state-dependent arrival rates are considered. Two recursion formulas for the Laplace–Stieltjes transforms of the busy periods under the FCFS and preempt resume LCFS service disciplines are obtained. It is shown that the busy-period distributions for the two service disciplines are, in general, different, in contrast to the fact that they coincide for ordinary M/G/1 queues. For deterministic service times and arrival rates non-increasing in the number of customers in the system, stochastic ordering between these two busy periods is also established.
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Shanthikumar, J. George, and Ushio Sumita. "On the busy-period distributions of M/G/1/K queues by state-dependent arrivals and FCFS/LCFS-P service disciplines." Journal of Applied Probability 22, no. 04 (1985): 912–19. http://dx.doi.org/10.1017/s0021900200108149.
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The busy-period distributions of M/G/1/K queues with state-dependent arrival rates are considered. Two recursion formulas for the Laplace–Stieltjes transforms of the busy periods under the FCFS and preempt resume LCFS service disciplines are obtained. It is shown that the busy-period distributions for the two service disciplines are, in general, different, in contrast to the fact that they coincide for ordinary M/G/1 queues. For deterministic service times and arrival rates non-increasing in the number of customers in the system, stochastic ordering between these two busy periods is also established.
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Liu, Lei, Kangjing Li, Pengfei Du, Fan Jiang, Xuewei Zhang, and Qi Han. "Age of Information in NOMA-IoT Networks: A Temporal Queuing Model Perspective." Mathematics 12, no. 10 (2024): 1440. http://dx.doi.org/10.3390/math12101440.
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The Internet of Things (IoT) with non-orthogonal multiple access (NOMA) has been anticipated to offer diverse real-time applications, wherein the crux is to guarantee the age of information (AoI) for dynamic traffic. However, the traffic temporal variation provokes the interdependence between queue status and interference, in which context the AoI performance remains to be further explored. In this paper, an analytical framework is established to characterize the AoI performance in NOMA-IoT networks with random Bernoulli and deterministic periodic arrivals. Particularly, a numerical algorithm is devised to obtain the queue service rate, and tractable expressions for AoI violation probability and average AoI under both the first-come first-served (FCFS) and the preemptive last-come first-served (LCFS-PR) service disciplines are derived. Simulations are conducted to validate the proposed analysis. The results unveil that LCFS-PR conduces to better AoI performance than FCFS, and yet the gain diverges for each device with different traffic arrival configurations. In addition, the result shows that with sporadic traffic arrival, the periodic pattern outperforms the Bernoulli pattern, whereas this advantage gradually diminishes with more frequent packet arrival.
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HE, Qi-Ming, and Attahiru Sule Alfa. "Computational analysis ofMMAP[K]/PH[K]/1 queues with a mixed FCFS and LCFS service discipline." Naval Research Logistics 47, no. 5 (2000): 399–421. http://dx.doi.org/10.1002/1520-6750(200008)47:5<399::aid-nav3>3.0.co;2-9.
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Robinson, Lawrence W., and Kevin B. Hendricks. "Using state-dependent processing rates to emulate SPT queue discipline in an FCPS queueing network." IIE Transactions 27, no. 4 (1995): 530–41. http://dx.doi.org/10.1080/07408179508936769.
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Shin, Yang Woo, and Bong Dae Choi. "A QUEUE WITH POSITIVE AND NEGATIVE ARRIVALS GOVERNED BY A MARKOV CHAIN." Probability in the Engineering and Informational Sciences 17, no. 4 (2003): 487–501. http://dx.doi.org/10.1017/s0269964803174049.
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We consider a single-server queue with exponential service time and two types of arrivals: positive and negative. Positive customers are regular ones who form a queue and a negative arrival has the effect of removing a positive customer in the system. In many applications, it might be more appropriate to assume the dependence between positive arrival and negative arrival. In order to reflect the dependence, we assume that the positive arrivals and negative arrivals are governed by a finite-state Markov chain with two absorbing states, say 0 and 0′. The epoch of absorption to the states 0 and 0′ corresponds to an arrival of positive and negative customers, respectively. The Markov chain is then instantly restarted in a transient state, where the selection of the new state is allowed to depend on the state from which absorption occurred.The Laplace–Stieltjes transforms (LSTs) of the sojourn time distribution of a customer, jointly with the probability that the customer completes his service without being removed, are derived under the combinations of service disciplines FCFS and LCFS and the removal strategies RCE and RCH. The service distribution of phase type is also considered.
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Seshadri, Sridhar, and Michael Pinedo. "Bounds on the Mean Delay in Multiclass Queueing Networks under Shortfall-Based Priority Rules." Probability in the Engineering and Informational Sciences 12, no. 3 (1998): 329–50. http://dx.doi.org/10.1017/s0269964800005234.
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A significant amount of recent research has been focused on the stability of multiclass open networks of queues (MONQs). It has been shown that these networks may be unstable under various queueing disciplines even when at each one of the nodes the arrival rate is less than the service rate. Clearly, in such cases the expected delay and the expected number of customers in the system are infinite. In this paper we propose a new class of scheduling rules that can be used in multiclass queueing networks. We refer to this class as the stable shortfall-based priority (SSBP) rules. This SSBP class itself belongs to a larger class of rules, which we refer to as the shortfall-based priority (SBP) rules. SBP is a generalization of the standard non-preemptive priority rule in which customers of the same priority class are served first-come, first-served (FCFS). Rules from SBP can mimic FCFS as well as the so-called strict or head-of-the-line priority disciplines. We show that the use of any rule from the SSBP class ensures stability in a broad class of MONQs found in practice. We proceed with the construction of a sample path inequality for the work done by an SSBP server and show how this inequality can be used to derive upper bounds for the delay when service times are bounded. Bounds for the expected delay of each class of customers in an MONQ are then obtained under the assumptions that the external arrival processes have i.i.d. interarrival times, the routings are deterministic and the service times at each step of the route are bounded. In order to derive these bounds for the average delay in an MONQ we make use of some of the classical ideas of Kingman.