The general N server finite queue

1971 ◽  
Vol 8 (04) ◽  
pp. 828-834 ◽  
Author(s):  
Asha Seth Kapadia

Kingman (1962) studied the effect of queue discipline on the mean and variance of the waiting time. He made no assumptions regarding the stochastic nature of the input and the service distributions, except that the input and service processes are independent of each other. When the following two conditions hold: (a) no server sits idle while there are customers waiting to be served; (b) the busy period is finite with probability one (i.e., the queue empties infinitely often with probability one); he has shown that the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in order of their arrival. Conditions (a) and (b) will henceforward be called Kingman conditions and a queueing system satisfying Kingman conditions will be referred to in the text as a Kingman queue.

1971 ◽  
Vol 8 (4) ◽  
pp. 828-834 ◽  
Author(s):  
Asha Seth Kapadia

Kingman (1962) studied the effect of queue discipline on the mean and variance of the waiting time. He made no assumptions regarding the stochastic nature of the input and the service distributions, except that the input and service processes are independent of each other. When the following two conditions hold: (a)no server sits idle while there are customers waiting to be served;(b)the busy period is finite with probability one (i.e., the queue empties infinitely often with probability one); he has shown that the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in order of their arrival. Conditions (a) and (b) will henceforward be called Kingman conditions and a queueing system satisfying Kingman conditions will be referred to in the text as a Kingman queue.


1968 ◽  
Vol 5 (3) ◽  
pp. 702-703 ◽  
Author(s):  
D. G. Tambouratzis

In this note, we consider a queueing system under any discipline which does not affect the distribution of the number of customers in the queue at any time. We shall show that the variance of the waiting time is a maximum when the queue discipline is “last come, first served”. This result complements that of Kingman [1] who showed that, under the same assumptions, the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in the order of their arrival.


1968 ◽  
Vol 5 (03) ◽  
pp. 702-703 ◽  
Author(s):  
D. G. Tambouratzis

In this note, we consider a queueing system under any discipline which does not affect the distribution of the number of customers in the queue at any time. We shall show that the variance of the waiting time is a maximum when the queue discipline is “last come, first served”. This result complements that of Kingman [1] who showed that, under the same assumptions, the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in the order of their arrival.


1969 ◽  
Vol 6 (1) ◽  
pp. 154-161 ◽  
Author(s):  
E.G. Enns

In the study of the busy period for a single server queueing system, three variables that have been investigated individually or at most in pairs are:1.The duration of the busy period.2.The number of customers served during the busy period.3.The maximum number of customers in the queue during the busy period.


2018 ◽  
Vol 103 (2) ◽  
pp. e1.27-e1
Author(s):  
Wheeler Lucy ◽  
Lewis Anthony

AimTo reduce the amount of time that short stay surgical patients wait for their discharge medication.MethodAn audit was carried out during 1 week in September 2016 of all discharge prescriptions written on the paediatric surgical ward. It measured time taken for the prescription to be processed by pharmacy and the number of prescriptions for paracetamol and ibuprofen.An intervention was introduced in February 2017: Parents/carers of children coming in for elective surgery were instructed (as part of their pre-admission letter) to buy paracetamol and ibuprofen. Nursing staff also reminded parents/carers when the child was admitted, that they would need to have these medicines on discharge.When a discharge letter was written, patients were not given paracetamol and ibuprofen. They were given a dosing instruction letter completed by the pharmacist at point of clinical check.Provision was made to supply the medication if a parent/carer could not provide it.The letter and dosing were designed by a paediatric pharmacist in conjunction with a paediatric anaesthetist to ensure patients were discharged on optimal pain control.1A re-audit was carried out over the course of 1 week in June 2017.ResultsIn September 2016, a total of 37 prescriptions were written from the paediatric surgery ward over a 1 week period. The average (mean) time to dispense a prescription was 76 min (range 16–200 min).In June 2017, a total of 30 prescriptions were written over a 1 week period. The average (mean) time to dispense a prescription was 51 min (range 10–124 min).There was a reduction in the mean waiting time of 25 min but this did not reach statistical significance (p=0.3).In September 2016, 12 prescriptions (32%) were for elective patients for paracetamol and ibuprofen only. Average time to dispense these prescriptions was 70 min.In June 2017, 7 prescriptions (23%) were for elective patients for paracetamol and ibuprofen only and an instruction letter was issued on the ward. Average time to complete these prescriptions was 21 min. There was a reduction in the mean waiting time of 49 min which was statistically significant (p=0.001). Time was measured from the time the pharmacist was informed the prescription was ready, to the time the medication or letter was ready and on the ward.ConclusionIntroduction of an instruction letter given to elective surgical patients on discharge appeared to lead to a shorter wait for discharge medication, although this was not statistically significant.A significant reduction in waiting time was seen for patients who were only prescribed paracetamol and ibuprofen and could be discharged without their prescription leaving the ward.Further work will look at trying to dispense/prepare more prescriptions at ward level, as this seems to be the most efficient system.It was noted that the number of prescriptions was slightly reduced (30 for the re-audit and 37 for the initial audit). The number of patients admitted and having surgery had remained constant.ReferenceAssociation of Paediatric Anaesthetists of Great Britain and Ireland. Good practice in postoperative and procedural pain management (2nd ed.). Paediatr Anaesth2012;22(Suppl. 1):S1–79.


2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
Yutae Lee ◽  
Bong Dae Choi ◽  
Bara Kim ◽  
Dan Keun Sung

This paper considers anM/G/1/Kqueueing system with push-out scheme which is one of the loss priority controls at a multiplexer in communication networks. The loss probability for the model with push-out scheme has been analyzed, but the waiting times are not available for the model. Using a set of recursive equations, this paper derives the Laplace-Stieltjes transforms (LSTs) of the waiting time and the push-out time of low-priority messages. These results are then utilized to derive the loss probability of each traffic type and the mean waiting time of high-priority messages. Finally, some numerical examples are provided.


1969 ◽  
Vol 6 (01) ◽  
pp. 154-161 ◽  
Author(s):  
E.G. Enns

In the study of the busy period for a single server queueing system, three variables that have been investigated individually or at most in pairs are: 1. The duration of the busy period. 2. The number of customers served during the busy period. 3. The maximum number of customers in the queue during the busy period.


2008 ◽  
Vol 2008 ◽  
pp. 1-20 ◽  
Author(s):  
B. Krishna Kumar ◽  
R. Rukmani ◽  
V. Thangaraj

We consider a two-heterogeneous-server queueing system with Bernoulli vacation in which customers arrive according to a Markovian arrival process (MAP). Servers returning from vacation immediately take another vacation if no customer is waiting. Using matrix-geometric method, the steady-state probability of the number of customers in the system is investigated. Some important performance measures are obtained. The waiting time distribution and the mean waiting time are also discussed. Finally, some numerical illustrations are provided.


2002 ◽  
Vol 18 (3) ◽  
pp. 611-618
Author(s):  
Markus Torkki ◽  
Miika Linna ◽  
Seppo Seitsalo ◽  
Pekka Paavolainen

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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