scholarly journals Two Coupled Queues with Vastly Different Arrival Rates: Critical Loading Case

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Charles Knessl ◽  
John A. Morrison
Keyword(s):  
Customer Service ◽  
Arrival Rate ◽  
Service Discipline ◽  
Processor Sharing ◽  
Steady State Probability ◽  
Poisson Arrival ◽  
Loading Case ◽  
Queue Lengths ◽  
Number Of Customers ◽  
Comparable Magnitude

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

scholarly journals Some Time-Dependent Properties of Symmetric M/G/1 Queues

2005 ◽  
Vol 42 (01) ◽  
pp. 223-234 ◽  
Author(s):  
Offer Kella ◽  
Bert Zwart ◽  
Onno Boxma
Keyword(s):  
Queue Length ◽  
Marginal Distribution ◽  
Geometric Distribution ◽  
Time Dependent ◽  
Service Discipline ◽  
Processor Sharing ◽  
Tail Behavior ◽  
Exponential Time ◽  
Preemptive Resume ◽  
Number Of Customers

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

scholarly journals Asymptotic Analysis of a Loss Model with Trunk Reservation I: Trunks Reserved for Fast Traffic

2008 ◽  
Vol 2008 ◽  
pp. 1-34 ◽  
Author(s):  
John A. Morrison ◽  
Charles Knessl
Keyword(s):  
Steady State ◽  
Asymptotic Analysis ◽  
Arrival Rate ◽  
Asymptotic Approximations ◽  
Steady State Probability ◽  
Total Load ◽  
Switched Network ◽  
Poisson Arrival ◽  
Blocking Probabilities ◽  
Single Link

We consider a model for a single link in a circuit-switched network. The link has C circuits, and the input consists of offered calls of two types, that we call primary and secondary traffic. Of the C links, R are reserved for primary traffic. We assume that both traffic types arrive as Poisson arrival streams. Assuming that C is large and R=O(1), the arrival rate of primary traffic is O(C), while that of secondary traffic is smaller, of the order O(C). The holding times of the primary calls are assumed to be exponentially distributed with unit mean. Those of the secondary calls are exponentially distributed with a large mean, that is, O(C). Thus, the primary calls have fast arrivals and fast service, compared to the secondary calls. The loads for both traffic types are comparable (O(C)), and we assume that the system is “critically loaded”; that is, the system's capacity is approximately equal to the total load. We analyze asymptotically the steady state probability that n1 (resp., n2) circuits are occupied by primary (resp., secondary) calls. In particular, we obtain two-term asymptotic approximations to the blocking probabilities for both traffic types.

scholarly journals TheN×D-BMAP/G/1 Queueing Model: Queue Contents and Delay Analysis

2011 ◽  
Vol 2011 ◽  
pp. 1-31 ◽  
Author(s):  
Bart Steyaert ◽  
Joris Walraevens ◽  
Dieter Fiems ◽  
Herwig Bruneel
Keyword(s):  
Closed Form ◽  
Customer Service ◽  
Probability Generating Function ◽  
Mean Value ◽  
Arrival Process ◽  
Single Server ◽  
Steady State Probability ◽  
Absolute Minimum ◽  
Number Of Customers ◽  
Customer Delay

We consider a single-server discrete-time queueing system with N sources, where each source is modelled as a correlated Markovian customer arrival process, and the customer service times are generally distributed. We focus on the analysis of the number of customers in the queue, the amount of work in the queue, and the customer delay. For each of these quantities, we will derive an expression for their steady-state probability generating function, and from these results, we derive closed-form expressions for key performance measures such as their mean value, variance, and tail distribution. A lot of emphasis is put on finding closed-form expressions for these quantities that reduce all numerical calculations to an absolute minimum.

scholarly journals Some Time-Dependent Properties of Symmetric M/G/1 Queues

2005 ◽  
Vol 42 (1) ◽  
pp. 223-234 ◽  
Author(s):  
Offer Kella ◽  
Bert Zwart ◽  
Onno Boxma
Keyword(s):  
Queue Length ◽  
Marginal Distribution ◽  
Geometric Distribution ◽  
Time Dependent ◽  
Service Discipline ◽  
Processor Sharing ◽  
Tail Behavior ◽  
Exponential Time ◽  
Preemptive Resume ◽  
Number Of Customers

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

Tollbooth tandem queues with infinite homogeneous servers

2015 ◽  
Vol 52 (4) ◽  
pp. 941-961 ◽  
Author(s):  
Xiuli Chao ◽  
Qi-Ming He ◽  
Sheldon Ross
Keyword(s):  
Performance Measures ◽  
System Performance ◽  
Stochastic Ordering ◽  
Tandem Queues ◽  
Poisson Arrival ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Steady State Solutions ◽  
Transient And Steady State

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

Hitting time in an M/G/1 queue

1999 ◽  
Vol 36 (03) ◽  
pp. 934-940 ◽  
Author(s):  
Sheldon M. Ross ◽  
Sridhar Seshadri
Keyword(s):  
Service Time ◽  
Time Distribution ◽  
Arrival Rate ◽  
Hitting Time ◽  
Service Time Distribution ◽  
Expected Time ◽  
Efficient Simulation ◽  
Simulation Procedure ◽  
Poisson Arrival ◽  
Stationary Delay

We study the expected time for the work in an M/G/1 system to exceed the level x, given that it started out initially empty, and show that it can be expressed solely in terms of the Poisson arrival rate, the service time distribution and the stationary delay distribution of the M/G/1 system. We use this result to construct an efficient simulation procedure.

scholarly journals A M/M/1 Queueing Network with Classical Retrial Policy

2021 ◽  
Vol 12 (1S) ◽  
pp. 329-337
Author(s):  
S. Shanmugasundaram, Et. al.
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Queueing Network ◽  
State Probability ◽  
Queueing Model ◽  
Steady State Probability ◽  
Numerical Examples ◽  
System Length ◽  
Number Of Customers ◽  
Little's Formula

In this paper we study the M/M/1 queueing model with retrial on network. We derive the steady state probability of customers in the network, the average number of customers in the all the three nodes in the system, the queue length, system length using little’s formula. The particular case is derived (no retrial). The numerical examples are given to test the correctness of the model.

scholarly journals Diffusion Limit of Multi-Server Retrial Queue with Setup Time

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2232
Author(s):  
Anatoly Nazarov ◽  
Alexander Moiseev ◽  
Tuan Phung-Duc ◽  
Svetlana Paul
Keyword(s):  
Asymptotic Methods ◽  
Retrial Queue ◽  
Power Saving ◽  
Setup Time ◽  
Diffusion Limit ◽  
Steady State Probability ◽  
Retrial Queueing System ◽  
Limiting Condition ◽  
Number Of Customers ◽  
Multi Server

In the paper, we consider a multi-server retrial queueing system with setup time which is motivated by applications in power-saving data centers with the ON-OFF policy, where an idle server is immediately turned off and an off server is set up upon arrival of a customer. Customers that find all the servers busy join the orbit and retry for service after an exponentially distributed time. For this model, we derive the stability condition which depends on the setup time and turns out to be more strict than that of the corresponding model with an infinite buffer which is independent of the setup time. We propose asymptotic methods to analyze the system under the condition that the delay in the orbit is extremely long. We show that the scaled-number of customers in the orbit converges to a diffusion process. Using this diffusion limit, we obtain approximations for the steady-state probability distribution of the number of busy servers and that of the number of customers in the orbit. We verify the accuracy of the approximations by simulations and numerical analysis. Numerical results show that the retrial system under the limiting condition consumes more energy than that with an infinite buffer in front of the servers.

scholarly journals Customer complaints and service policy in electronic commerce

2013 ◽  
Vol 44 (3) ◽  
pp. 15-20 ◽  
Author(s):  
Y-W. Fan ◽  
Y-F. Miao ◽  
S-C. Wu
Keyword(s):  
Electronic Commerce ◽  
Customer Service ◽  
Service Failure ◽  
Business Managers ◽  
Customer Complaints ◽  
Prevention Policies ◽  
Customer Complaint ◽  
Service Policy ◽  
Number Of Customers

Handling customer complaints is an important strategy to retain customers. Therefore, in the event of service failure, e-retailers should concentrate on recovery policies. However, studies discussing prevention policies to avoid customer complaints are scant. This study collected 5933 real customer complaint data from an electronic commerce customer-service database and classified customer complaints into 6 types. The findings showed that a number of customers were dishonest and took advantage of recovery policies. After interviewing business managers and consultants, this research suggests that e-retailers have prevention policies to guarantee accuracy of packaging and delivery processes. Prevention policies can reduce customer complaints, and avoid extra costs for businesses conducting recovery policies.

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