Work-modulated queues with applications to storage processes

1992 ◽  
Vol 29 (03) ◽  
pp. 699-712 ◽  
Author(s):  
Sid Browne ◽  
Karl Sigman
Keyword(s):  
Sufficient Conditions ◽  
Arrival Process ◽  
Service Rate ◽  
Single Server ◽  
Steady State Distribution ◽  
State Distribution ◽  
Deterministic Function ◽  
Customer Delay ◽  
Finite Moments ◽  
Storage Processes

We study two FIFO single-server queueing models in which both the arrival and service processes are modulated by the amount of work in the system. In the first model, the nth customer's service time, Sn , depends upon their delay, Dn , in a general Markovian way and the arrival process is a non-stationary Poisson process (NSPP) modulated by work, that is, with an intensity that is a general deterministic function g of work in system V(t). Some examples are provided. In our second model, the arrivals once again form a work-modulated NSPP, but, each customer brings a job consisting of an amount of work to be processed that is i.i.d. and the service rate is a general deterministic function r of work. This model can be viewed as a storage (dam) model (Brockwell et al. (1982)), but, unlike previous related literature, (where the input is assumed work-independent and stationary), we allow a work-modulated NSPP. Our approach involves an elementary use of Foster's criterion (via Tweedie (1976)) and in addition to obtaining new results, we obtain new and simplified proofs of stability for some known models. Using further criteria of Tweedie, we establish sufficient conditions for the steady-state distribution of customer delay and sojourn time to have finite moments.

Work-modulated queues with applications to storage processes

1992 ◽  
Vol 29 (3) ◽  
pp. 699-712 ◽  
Author(s):  
Sid Browne ◽  
Karl Sigman
Keyword(s):  
Sufficient Conditions ◽  
Arrival Process ◽  
Service Rate ◽  
Single Server ◽  
Steady State Distribution ◽  
State Distribution ◽  
Deterministic Function ◽  
Customer Delay ◽  
Finite Moments ◽  
Storage Processes

We study two FIFO single-server queueing models in which both the arrival and service processes are modulated by the amount of work in the system. In the first model, the nth customer's service time, Sn, depends upon their delay, Dn, in a general Markovian way and the arrival process is a non-stationary Poisson process (NSPP) modulated by work, that is, with an intensity that is a general deterministic function g of work in system V(t). Some examples are provided. In our second model, the arrivals once again form a work-modulated NSPP, but, each customer brings a job consisting of an amount of work to be processed that is i.i.d. and the service rate is a general deterministic function r of work. This model can be viewed as a storage (dam) model (Brockwell et al. (1982)), but, unlike previous related literature, (where the input is assumed work-independent and stationary), we allow a work-modulated NSPP. Our approach involves an elementary use of Foster's criterion (via Tweedie (1976)) and in addition to obtaining new results, we obtain new and simplified proofs of stability for some known models. Using further criteria of Tweedie, we establish sufficient conditions for the steady-state distribution of customer delay and sojourn time to have finite moments.

BALKING AND RENEGING IN M/G/s SYSTEMS EXACT ANALYSIS AND APPROXIMATIONS

2008 ◽  
Vol 22 (3) ◽  
pp. 355-371 ◽  
Author(s):  
Liqiang Liu ◽  
Vidyadhar G. Kulkarni
Keyword(s):  
Steady State ◽  
Single Server ◽  
Impatient Customers ◽  
Operating Characteristics ◽  
Steady State Distribution ◽  
State Distribution ◽  
Numerical Examples ◽  
Exact Analysis ◽  
Analytical Results ◽  
Server System

We consider the virtual queuing time (vqt, also known as work-in-system, or virtual-delay) process in an M/G/s queue with impatient customers. We focus on the vqt-based balking model and relate it to reneging behavior of impatient customers in terms of the steady-state distribution of the vqt process. We construct a single-server system, analyze its operating characteristics, and use them to approximate the multiserver system. We give both analytical results and numerical examples. We conduct simulation to assess the accuracy of the approximation.

On stability of state-dependent queues and acyclic queueing networks

1989 ◽  
Vol 21 (3) ◽  
pp. 681-701 ◽  
Author(s):  
Nicholas Bambos ◽  
Jean Walrand
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Service Rate ◽  
Single Server ◽  
General Function ◽  
State Dependent ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Periodic Inputs ◽  
Networks Of Queues

We consider a single server first-come-first-served queue with a stationary and ergodic input. The service rate is a general function of the workload in the queue. We provide the necessary and sufficient conditions for the stability of the system and the asymptotic convergence of the workload process to a finite stationary process at large times. Then, we consider acyclic networks of queues in which the service rate of any queue is a function of the workloads of this and of all the preceding queues. The stability problem is again studied. The results are then extended to analogous systems with periodic inputs.

Useful martingales for stochastic storage processes with Lévy input

1992 ◽  
Vol 29 (2) ◽  
pp. 396-403 ◽  
Author(s):  
Offer Kella ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Lévy Process ◽  
Finite Interval ◽  
Levy Process ◽  
Secondary Process ◽  
Stochastic Integration ◽  
Steady State Distribution ◽  
State Distribution ◽  
Storage Network ◽  
Storage Processes

We apply the general theory of stochastic integration to identify a martingale associated with a Lévy process modified by the addition of a secondary process of bounded variation on every finite interval. This martingale can be applied to queues and related stochastic storage models driven by a Lévy process. For example, we have applied this martingale to derive the (non-product-form) steady-state distribution of a two-node tandem storage network with Lévy input and deterministic linear fluid flow out of the nodes.

scholarly journals Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process: D-$MAP/G_n^{(a,b)}/1$

2021 ◽  
Author(s):  
Umesh Chandra Gupta ◽  
Nitin Kumar ◽  
Sourav Pradhan ◽  
Farida Parvez Barbhuiya ◽  
Mohan L Chaudhry
Keyword(s):  
Generating Function ◽  
Discrete Time ◽  
Service Time ◽  
Batch Size ◽  
Arrival Process ◽  
General Distribution ◽  
Complete Analysis ◽  
Service Rate ◽  
Single Server ◽  
Digital Platforms

Discrete-time queueing models find a large number of applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems and computer networks. In this paper, we analyze an infinite-buffer queueing model with discrete Markovian arrival process. The units on arrival are served in batches by a single server according to the general bulk-service rule, and the service time follows general distribution with service rate depending on the size of the batch being served. We mathematically formulate the model using the supplementary variable technique and obtain the vector generating function at the departure epoch. The generating function is in turn used to extract the joint distribution of queue and server content in terms of the roots of the characteristic equation. Further, we develop the relationship between the distribution at the departure epoch and the distribution at arbitrary, pre-arrival and outside observer's epochs, where the first is used to obtain the latter ones. We evaluate some essential performance measures of the system and also discuss the computing process extensively which is demonstrated by some numerical examples.

On stability of state-dependent queues and acyclic queueing networks

1989 ◽  
Vol 21 (03) ◽  
pp. 681-701 ◽  
Author(s):  
Nicholas Bambos ◽  
Jean Walrand
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Service Rate ◽  
Single Server ◽  
General Function ◽  
State Dependent ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Periodic Inputs ◽  
Networks Of Queues

We consider a single server first-come-first-served queue with a stationary and ergodic input. The service rate is a general function of the workload in the queue. We provide the necessary and sufficient conditions for the stability of the system and the asymptotic convergence of the workload process to a finite stationary process at large times. Then, we consider acyclic networks of queues in which the service rate of any queue is a function of the workloads of this and of all the preceding queues. The stability problem is again studied. The results are then extended to analogous systems with periodic inputs.

Average delay in queues with non-stationary Poisson arrivals

1978 ◽  
Vol 15 (03) ◽  
pp. 602-609 ◽  
Author(s):  
Sheldon M. Ross
Keyword(s):  
Poisson Process ◽  
Intensity Function ◽  
Arrival Process ◽  
Single Server ◽  
Actual Process ◽  
Average Delay ◽  
Average Value ◽  
Poisson Arrival ◽  
Poisson Arrivals ◽  
Customer Delay

One of the major difficulties in attempting to apply known queueing theory results to real problems is that almost always these results assume a time-stationary Poisson arrival process, whereas in practice the actual process is almost invariably non-stationary. In this paper we consider single-server infinite-capacity queueing models in which the arrival process is a non-stationary process with an intensity function ∧(t), t ≧ 0, which is itself a random process. We suppose that the average value of the intensity function exists and is equal to some constant, call it λ, with probability 1. We make a conjecture to the effect that ‘the closer {∧(t), t ≧ 0} is to the stationary Poisson process with rate λ ' then the smaller is the average customer delay, and then we verify the conjecture in the special case where the arrival process is an interrupted Poisson process.

Useful martingales for stochastic storage processes with Lévy input

1992 ◽  
Vol 29 (02) ◽  
pp. 396-403 ◽  
Author(s):  
Offer Kella ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Lévy Process ◽  
Finite Interval ◽  
Levy Process ◽  
Secondary Process ◽  
Stochastic Integration ◽  
Steady State Distribution ◽  
State Distribution ◽  
Storage Network ◽  
Storage Processes

We apply the general theory of stochastic integration to identify a martingale associated with a Lévy process modified by the addition of a secondary process of bounded variation on every finite interval. This martingale can be applied to queues and related stochastic storage models driven by a Lévy process. For example, we have applied this martingale to derive the (non-product-form) steady-state distribution of a two-node tandem storage network with Lévy input and deterministic linear fluid flow out of the nodes.

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.

On a decomposition result for a class of vacation queueing systems

1990 ◽  
Vol 27 (1) ◽  
pp. 227-231 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Steady State ◽  
Time Distribution ◽  
Queueing Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
Steady State Distribution ◽  
State Distribution ◽  
Random Size ◽  
Poisson Arrivals ◽  
General Service

We consider a single-server infinite-capacity queueing sysem with Poisson arrivals of customer groups of random size and a general service time distribution, the server of which applies a general exhaustive service vacation policy. We are concerned with the steady-state distribution of the actual waiting time of a customer arriving while the server is active.

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