Multiple channel queues in heavy traffic. I

1970 ◽  
Vol 2 (1) ◽  
pp. 150-177 ◽  
Author(s):  
Donald L. Iglehart ◽  
Ward Whitt
Keyword(s):  
Heavy Traffic ◽  
Arrival Rate ◽  
Standard System ◽  
Interarrival Time ◽  
Service Rate ◽  
Service Channel ◽  
Multiple Channel ◽  
System A ◽  
The Mean ◽  
Service Channels

The queueing systems considered in this paper consist of r independent arrival channels and s independent service channels, where as usual the arrival and service channels are independent. Arriving customers form a single queue and are served in the order of their arrival without defections. We shall treat two distinct modes of operation for the service channels. In the standard system a waiting customer is assigned to the first available service channel and the servers (servers ≡ service channels) are shut off when they are idle. Thus the classical GI/G/s system is a special case of our standard system. In the modified system a waiting customer is assigned to the service channel that can complete his service first and the servers are not shut off when they are idle. While the modified system is of some interest in its own right, we introduce it primarily as an analytical tool. Let λi denote the arrival rate (reciprocal of the mean interarrival time) in the ith arrival channel and μj the service rate (reciprocal of the mean service time) in the jth service channel. Then is the total arrival rate to the system and is the maximum service rate of the system. As a measure of congestion we define the traffic intensity ρ = λ/μ.

Multiple channel queues in heavy traffic. I

1970 ◽  
Vol 2 (01) ◽  
pp. 150-177 ◽  
Author(s):  
Donald L. Iglehart ◽  
Ward Whitt
Keyword(s):  
Heavy Traffic ◽  
Arrival Rate ◽  
Standard System ◽  
Interarrival Time ◽  
Service Rate ◽  
Service Channel ◽  
Multiple Channel ◽  
System A ◽  
The Mean ◽  
Service Channels

The queueing systems considered in this paper consist of r independent arrival channels and s independent service channels, where as usual the arrival and service channels are independent. Arriving customers form a single queue and are served in the order of their arrival without defections. We shall treat two distinct modes of operation for the service channels. In the standard system a waiting customer is assigned to the first available service channel and the servers (servers ≡ service channels) are shut off when they are idle. Thus the classical GI/G/s system is a special case of our standard system. In the modified system a waiting customer is assigned to the service channel that can complete his service first and the servers are not shut off when they are idle. While the modified system is of some interest in its own right, we introduce it primarily as an analytical tool. Let λ i denote the arrival rate (reciprocal of the mean interarrival time) in the ith arrival channel and μ j the service rate (reciprocal of the mean service time) in the jth service channel. Then is the total arrival rate to the system and is the maximum service rate of the system. As a measure of congestion we define the traffic intensity ρ = λ/μ.

scholarly journals Performance Modelling of Health-care Service Delivery in Adekunle Ajasin University, Akungba-Akoko, Nigeria Using Queuing Theory

2020 ◽  
pp. 119-127
Author(s):  
Orimoloye Segun Michael
Keyword(s):  
Health Care ◽  
Service Delivery ◽  
Queuing Theory ◽  
Health Care Service ◽  
Health Centre ◽  
Arrival Rate ◽  
Queuing System ◽  
Service Rate ◽  
Suitable Model ◽  
The Mean

The queuing theory is the mathematical approach to the analysis of waiting lines in any setting where arrivals rate of the subject is faster than the system can handle. It is applicable to the health care setting where the systems have excess capacity to accommodate random variation. Therefore, the purpose of this study was to determine the waiting, arrival and service times of patients at AAUA Health- setting and to model a suitable queuing system by using simulation technique to validate the model. This study was conducted at AAUA Health- Centre Akungba Akoko. It employed analytical and simulation methods to develop a suitable model. The collection of waiting time for this study was based on the arrival rate and service rate of patients at the Outpatient Centre. The data was calculated and analyzed using Microsoft Excel. Based on the analyzed data, the queuing system of the patient current situation was modelled and simulated using the PYTHON software. The result obtained from the simulation model showed that the mean arrival rate of patients on Friday week1 was lesser than the mean service rate of patients (i.e. 5.33> 5.625 (λ > µ). What this means is that the waiting line would be formed which would increase indefinitely; the service facility would always be busy. The analysis of the entire system of the AAUA health centre showed that queue length increases when the system is very busy. This work therefore evaluated and predicted the system performance of AAUA Health-Centre in terms of service delivery and propose solutions on needed resources to improve the quality of service offered to the patients visiting this health centre.

Multiple channel queues in heavy traffic. III: random server selection

1970 ◽  
Vol 2 (2) ◽  
pp. 370-375 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Queueing System ◽  
Heavy Traffic ◽  
Multiple Channel ◽  
Server Selection ◽  
Selection For ◽  
Selection Probabilities ◽  
Service Channels

As in [4] and [5], we study service facilities with r arrival channels and s service channels. However, here we assume that customers, immediately upon arrival, randomly select one of the s service channels. Successive customers make this choice independently, choosing server i with probability pi, p1 + · · · + ps = 1. Customers are then served by the servers they select in order of their arrival without defections. The average processing rates as well as the server selection probabilities may vary from server to server, but again we assume the r arrival channels are independent and independent of the service channels. The service channels are not independent, however, because of the random server selection. For simplicity, we only consider a single queueing system; the extension to sequences follows immediately using the argument of [5].

Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (3) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the form λ(t) = λ(0) — βt2 for some constant β. Diffusion approximations show that for λ(0) sufficiently close to the service rate μ, the mean queue length at time 0 is proportional to β–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all λ(0) and β. Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

Multiple channel queues in heavy traffic. II: sequences, networks, and batches

1970 ◽  
Vol 2 (02) ◽  
pp. 355-369 ◽  
Author(s):  
Donald L. Iglehart ◽  
Ward Whitt
Keyword(s):  
Stochastic Processes ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
General Setting ◽  
Critical Value ◽  
Multiple Channel ◽  
Heavy Traffic Limit ◽  
Service Channels ◽  
Traffic Behavior

This paper is a sequel to [7], in which heavy traffic limit theorems were proved for various stochastic processes arising in a single queueing facility with r arrival channels and s service channels. Here we prove similar theorems for sequences of such queueing facilities. The same heavy traffic behavior prevails in many cases in this more general setting, but new heavy traffic behavior is observed when the sequence of traffic intensities associated with the sequence of queueing facilities approaches the critical value (ρ = 1) at appropriate rates.

Multiple channel queues in heavy traffic. II: sequences, networks, and batches

1970 ◽  
Vol 2 (2) ◽  
pp. 355-369 ◽  
Author(s):  
Donald L. Iglehart ◽  
Ward Whitt
Keyword(s):  
Stochastic Processes ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
General Setting ◽  
Critical Value ◽  
Multiple Channel ◽  
Heavy Traffic Limit ◽  
Service Channels ◽  
Traffic Behavior

This paper is a sequel to [7], in which heavy traffic limit theorems were proved for various stochastic processes arising in a single queueing facility with r arrival channels and s service channels. Here we prove similar theorems for sequences of such queueing facilities. The same heavy traffic behavior prevails in many cases in this more general setting, but new heavy traffic behavior is observed when the sequence of traffic intensities associated with the sequence of queueing facilities approaches the critical value (ρ = 1) at appropriate rates.

Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (03) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the formλ(t) =λ(0) —βt2for some constantβ.Diffusion approximations show that forλ(0) sufficiently close to the service rateμ, the mean queue length at time 0 is proportional toβ–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for allλ(0) andβ.Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

Multiple channel queues in heavy traffic. III: random server selection

1970 ◽  
Vol 2 (02) ◽  
pp. 370-375 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Queueing System ◽  
Heavy Traffic ◽  
Multiple Channel ◽  
Server Selection ◽  
Selection For ◽  
Selection Probabilities ◽  
Service Channels

As in [4] and [5], we study service facilities with r arrival channels and s service channels. However, here we assume that customers, immediately upon arrival, randomly select one of the s service channels. Successive customers make this choice independently, choosing server i with probability p i , p 1 + · · · + p s = 1. Customers are then served by the servers they select in order of their arrival without defections. The average processing rates as well as the server selection probabilities may vary from server to server, but again we assume the r arrival channels are independent and independent of the service channels. The service channels are not independent, however, because of the random server selection. For simplicity, we only consider a single queueing system; the extension to sequences follows immediately using the argument of [5].

Characterizing losses during busy periods in finite buffer systems

2003 ◽  
Vol 40 (1) ◽  
pp. 242-249 ◽  
Author(s):  
Erol A. Peköz ◽  
Rhonda Righter ◽  
Cathy H. Xia
Keyword(s):  
Busy Period ◽  
Arrival Rate ◽  
Batch Size ◽  
Buffer Size ◽  
Service Rate ◽  
Finite Buffer ◽  
Initial Number ◽  
Poisson Arrivals ◽  
The Mean ◽  
Number Of Customers

For multiple-server finite-buffer systems with batch Poisson arrivals, we explore how the distribution of the number of losses during a busy period changes with the buffer size and the initial number of customers. We show that when the arrival rate equals the maximal service rate (ρ= 1), as the buffer size increases the number of losses in a busy period increases in the convex sense, and whenρ> 1, as the buffer size increases the number of busy period losses increases in the increasing convex sense. Also, the number of busy period losses is stochastically increasing in the initial number of customers. A consequence of our results is that, whenρ= 1, the mean number of busy period losses equals the mean batch size of arrivals regardless of the buffer size. We show that this invariance does not extend to general arrival processes.

scholarly journals DELAYS AT SIGNALIZED INTERSECTIONS WITH EXHAUSTIVE TRAFFIC CONTROL

2012 ◽  
Vol 26 (3) ◽  
pp. 337-373 ◽  
Author(s):  
M.A.A. Boon ◽  
I.J.B.F. Adan ◽  
E.M.M. Winands ◽  
D.G. Down
Keyword(s):  
Closed Form ◽  
Traffic Control ◽  
Heavy Traffic ◽  
Interarrival Time ◽  
Traffic Signal Control ◽  
Stay Green ◽  
Traffic Lights ◽  
Traffic Intersection ◽  
Poisson Arrivals ◽  
The Mean

In this paper, we study a traffic intersection with vehicle-actuated traffic signal control. Traffic lights stay green until all lanes within a group are emptied. Assuming general renewal arrival processes, we derive exact limiting distributions of the delays under heavy traffic (HT) conditions. Furthermore, we derive the light traffic (LT) limit of the mean delays for intersections with Poisson arrivals, and develop a heuristic adaptation of this limit to capture the LT behavior for other interarrival-time distributions. We combine the LT and HT results to develop closed-form approximations for the mean delays of vehicles in each lane. These closed-form approximations are quite accurate, very insightful, and simple to implement.

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