PERFORMANCE ESTIMATION OFM/DK/1 QUEUE UNDER FAIR SOJOURN PROTOCOL IN HEAVY TRAFFIC

2008 ◽  
Vol 23 (1) ◽  
pp. 61-74
Author(s):  
Yingdong Lu
Keyword(s):  
Markov Process ◽  
Waiting Time ◽  
Service Time ◽  
Infinitesimal Generator ◽  
Heavy Traffic ◽  
Queuing System ◽  
Performance Estimation ◽  
Processor Sharing ◽  
Average Waiting Time ◽  
Processor Sharing Queues

We study the performance of aM/DK/1 queue under Fair Sojourn Protocol (FSP). We use a Markov process with mixed real- and measure-valued states to characterize the queuing process of system and its related processor sharing queue. The infinitesimal generator of the Markov process is derived. Classifying customers according to their service time, using techniques in multiclass queuing system, and borrowing recently developed heavy traffic results for processor-sharing queues, we are able to derive approximations for average waiting time for the jobs.

On a relationship between processor-sharing queues and Crump–Mode–Jagers branching processes

1992 ◽  
Vol 24 (3) ◽  
pp. 653-698 ◽  
Author(s):  
Sergei Grishechkin
Keyword(s):  
Processing Time ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Processor Sharing ◽  
Batch Arrivals ◽  
Processor Sharing Queues ◽  
Residual Processing

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.

scholarly journals Small Queuing Restaurant Sustainable Revenue Management

2020 ◽  
Vol 12 (8) ◽  
pp. 3477
Author(s):  
Kwangji Kim ◽  
Mi-Jung Kim ◽  
Jae-Kyoon Jun
Keyword(s):  
South Korea ◽  
Revenue Management ◽  
Waiting Time ◽  
Equilibrium Points ◽  
Management Strategies ◽  
Queuing System ◽  
Utilization Rate ◽  
Trade Off ◽  
Average Waiting Time ◽  
Price Increases

When competitive small restaurants have queues in peak periods, they lack strategies to cope. However, few studies have examined small restaurants’ revenue management strategies at peak times. This research examines how such small restaurants in South Korea can improve their profitability by adapting their price increases, table mix, and the equilibrium points of the utilization rates, and reports the following findings based on the analysis of two studies. In Study 1, improving profitability by increasing prices should carefully consider the magnitude and timing. In Study 2, when implementing the table mix strategy, seat occupancy and profit also increase, and we further find the equilibrium points of the utilization rates. Under a queuing system, the utilization rate and average waiting time are also identified as having a trade-off relationship. The results provide insights into how managers of small restaurants with queues can develop efficient revenue management strategies to manage peak hours.

Development of a Heuristics Model for Simplifying a Multi-Channel Queuing System

2011 ◽  
Vol 367 ◽  
pp. 647-652
Author(s):  
B. Kareem ◽  
A. A. Aderoba
Keyword(s):  
Waiting Time ◽  
Channel Model ◽  
Single Channel ◽  
Queuing System ◽  
Queuing Model ◽  
Average Waiting Time ◽  
Queuing Models ◽  
Significant Difference ◽  
Number Of Customers ◽  
Proportionality Principle

Queuing model has been discussed widely in literature. The structures of queuing systems are broadly divided into three namely; single, multi-channel, and mixed. Equations for solving these queuing problems vary in complexity. The most complex of them is the multi-channel queuing problem. A heuristically simplified equation based on relative comparison, using proportionality principle, of the measured effectiveness from the single and multi-channel models seems promising in solving this complex problem. In this study, six different queuing models were used from which five of them are single-channel systems while the balance is multi-channel. Equations for solving these models were identified based on their properties. Queuing models’ performance parameters were measured using relative proportionality principle from which complexity of multi-channel system was transformed to a simple linear relation of the form = . This showed that the performance obtained from single channel model has a linear relationship with corresponding to multi-channel, and is a factor which varies with the structure of queuing system. The model was tested with practical data collected on the arrival and departure of customers from a cocoa processing factory. The performances obtained based on average number of customers on line , average number of customers in the system , average waiting time in line and average waiting time in the system, under certain conditions showed no significant difference between using heuristics and analytical models.

A queue with Markov-dependent service times

1968 ◽  
Vol 5 (02) ◽  
pp. 461-466
Author(s):  
Gerold Pestalozzi
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Single Channel ◽  
Probability Generating Function ◽  
Average Waiting Time ◽  
Poisson Input ◽  
The Mean

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

scholarly journals Spectral expansion method for analysis of a system with shifted Erlang and hyper-Erlang distributions

2021 ◽  
Vol 24 (2) ◽  
pp. 55-61
Author(s):  
Veniamin N. Tarasov ◽  
Nadezhda F. Bakhareva
Keyword(s):  
Waiting Time ◽  
Queuing Theory ◽  
Expansion Method ◽  
Spectral Expansion ◽  
Queuing System ◽  
Erlang Distribution ◽  
Modern Theory ◽  
Calculation Formula ◽  
Average Waiting Time ◽  
Erlang Distributions

In this paper, we obtained a spectral expansion of the solution to the Lindley integral equation for a queuing system with a shifted Erlang input flow of customers and a hyper-Erlang distribution of the service time. On its basis, a calculation formula is derived for the average waiting time in the queue for this system in a closed form. As you know, all other characteristics of the queuing system are derivatives of the average waiting time. The resulting calculation formula complements and expands the well-known unfinished formula for the average waiting time in queue in queuing theory for G/G/1 systems. In the theory of queuing, studies of private systems of the G/G/1 type are relevant due to the fact that they are actively used in the modern theory of teletraffic, as well as in the design and modeling of various data transmission systems.

GI/G/1 processor sharing queue in heavy traffic

1994 ◽  
Vol 26 (2) ◽  
pp. 539-555 ◽  
Author(s):  
Sergei Grishechkin
Keyword(s):  
Queue Length ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Processor Sharing ◽  
Traffic Intensity ◽  
Random Measures ◽  
Processing Times ◽  
Processor Sharing Queues ◽  
Limiting Behaviour ◽  
Residual Processing

Consider GI/G/1 processor sharing queues with traffic intensity tending to 1. Using the theory of random measures and the theory of branching processes we investigate the limiting behaviour of the queue length, sojourn time and random measures describing attained and residual processing times of customers present.

scholarly journals Heavy-traffic limits for polling models with exhaustive service and non-FCFS service order policies

2015 ◽  
Vol 47 (04) ◽  
pp. 989-1014 ◽  
Author(s):  
P. Vis ◽  
R. Bekker ◽  
R. D. van der Mei
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Processor Sharing ◽  
Waiting Time Distribution ◽  
Asymptotic Results ◽  
Local Service ◽  
Waiting Time Distributions ◽  
Polling Models ◽  
The Impact

We study cyclic polling models with exhaustive service at each queue under a variety of non-FCFS (first-come-first-served) local service orders, namely last-come-first-served with and without preemption, random-order-of-service, processor sharing, the multi-class priority scheduling with and without preemption, shortest-job-first, and the shortest remaining processing time policy. For each of these policies, we first express the waiting-time distributions in terms of intervisit-time distributions. Next, we use these expressions to derive the asymptotic waiting-time distributions under heavy-traffic assumptions, i.e. when the system tends to saturate. The results show that in all cases the asymptotic waiting-time distribution at queueiis fully characterized and of the form Γ Θi, with Γ and Θiindependent, and where Γ is gamma distributed with known parameters (and the same for all scheduling policies). We derive the distribution of the random variable Θiwhich explicitly expresses the impact of the local service order on the asymptotic waiting-time distribution. The results provide new fundamental insight into the impact of the local scheduling policy on the performance of a general class of polling models. The asymptotic results suggest simple closed-form approximations for the complete waiting-time distributions for stable systems with arbitrary load values.

scholarly journals Deskripsi Sistem Antrian pada Klinik Dokter Spesialis Penyakit Dalam

2012 ◽  
Vol 12 (1) ◽  
pp. 72
Author(s):  
Deiby T Salaki
Keyword(s):  
Direct Observation ◽  
Waiting Time ◽  
Arrival Rate ◽  
Queuing System ◽  
Queuing Model ◽  
Average Waiting Time ◽  
Number Of Patients

DESKRIPSI SISTEM ANTRIAN PADA KLINIK DOKTER SPESIALIS PENYAKIT DALAM ABSTRAK Penelitian ini dilakukan untuk mengetahui deskripsi sistem antrian pada klinik dokter internist. Pengumpulan data dilakukan secara langsung pada klinik dokter internist JHA selama 12 hari, selama 2 jam waktu pengamatan tiap harinya pada periode sibuk.. Model antrian yang digunakan adalah model (M/M/1) : (FIFO/~/~), tingkat kedatangan bersebaran poisson, waktu pelayanan bersebaran eksponensial, dengan jumlah pelayanan adalah seorang dokter, disiplin antrian yang digunakan adalah pasien yang pertama datang yang pertama dilayani, jumlah pelayanan dalam sistem dan ukuran populasi pada sumber masukan adalah tak berhingga.  Sistem antrian pada klinik ini memiliki kecepatan kedatangan pelayanan anamnesa rata-rata  menit 1 orang pasien datang, kecepatan kedatangan pelayanan pemeriksaan fisik rata-rata  menit 1 orang pasien datang, rata-rata waktu pelayanan anamnesa untuk  seorang pasien  menit, rata-rata waktu pelayanan pemeriksaan fisik untuk  seorang pasien  menit, peluang kesibukan  pelayanan anamnesa sebesar , peluang kesibukan  pelayanan pemeriksaan fisik sebesar , dan peluang pelayanan anamnesa menganggur sebesar , peluang pelayanan pemeriksaan fisik menganggur sebesar . Rata-rata banyaknya pengantri untuk anamnesa adalah  pasien sedangkan untuk pemeriksaan fisik  pasien, rata-rata banyaknya pengantri dalam sistem adalah  pasien, waktu rata-rata seorang pasien dalam klinik adalah  menit, waktu rata-rata seseorang pasien untuk antri adalah  menit. Kata kunci: Sistem Antrian, Klinik Penyakit Dalam  DESCRIPTION OF QUEUING SYSTEM AT THE INTERNIST CLINIC ABSTRACT This research determines the description of queuing system at the internist Clinic. Data collected by direct observation during 12 days and in 2 hours. Queuing model that used is model of (M/M/1): (FIFO /~/~). Based on the research, the clinic has 3.256 minutes per patient in average arrival rate for anamnesys, the average arrival rate for diaagnosys is 3.255 minutes per patient, average service speed for anamnesys is 2.675 minutes per patient, average service speed for diagnosys is 12.635 minutes, the probability of busy periods for anamnesys is 0.864, the probability of busy periods for diagnosys is 0.832 and probability of all free services or no patient in the anamnesys equal to 0.136, probability of all free services or no patient in the anamnesys equal to 0.168. The average number of patients in anamnesys queue is 5 patients, the average number of patients in diagnosys queue is 4 patients, the average number of patients in the system is 10 patients, the average waiting time in the system is 47.078 minutes and the average queuing time is 31.660 minutes. Keywords: Queuing system, internist clinic

scholarly journals COMPARISON OF TWO FORMS OF ERLANGIAN DISTRIBUTION LAW IN QUEUING THEORY

2021 ◽  
pp. 48-56
Author(s):  
V. N. Tarasov
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Waiting Time ◽  
Service Time ◽  
Queuing Theory ◽  
Time Shift ◽  
Queuing Systems ◽  
Average Waiting Time ◽  
Coefficients Of Variation ◽  
Erlang Distributions

Context. For modeling various data transmission systems, queuing systems G/G/1 are in demand, this is especially important because there is no final solution for them in the general case. The problem of the derivation in closed form of the solution for the average waiting time in the queue for ordinary system with erlangian input distributions of the second order and for the same system with shifted to the right distributions is considered. Objective. Obtaining a solution for the main system characteristic – the average waiting time for queue requirements for three types of queuing systems of type G/G/1 with usual and shifted erlangian input distributions. Method. To solve this problem, we used the classical method of spectral decomposition of the solution of Lindley integral equation, which allows one to obtain a solution for average the waiting time for systems under consideration in a closed form. For the practical application of the results obtained, the well-known method of moments of the theory of probability was used. Results. For the first time, spectral expansions of the solution of the Lindley integral equation for systems with ordinary and shifted Erlang distributions are obtained, with the help of which the calculation formulas for the average waiting time in the queue for the above systems in closed form are derived. Conclusions. The difference between the usual and normalized distribution is that the normalized distribution has a mathematical expectation independent of the order of the distribution k, therefore, the normalized and normal Erlang distributions differ in numerical characteristics. The introduction of the time shift parameter in the laws of input flow distribution and service time for the systems under consideration turns them into systems with a delay with a shorter waiting time. This is because the time shift operation reduces the coefficient of variation in the intervals between the receipts of the requirements and their service time, and as is known from queuing theory, the average wait time of requirements is related to these coefficients of variation by a quadratic dependence. The system with usual erlangian input distributions of the second order is applicable only at a certain point value of the coefficients of variation of the intervals between the receipts of the requirements and their service time. The same system with shifted distributions allows us to operate with interval values of coefficients of variations, which expands the scope of these systems. This approach allows us to calculate the average delay for these systems in mathematical packages for a wide range of traffic parameters.

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