scholarly journals The Single-Server Queue with the Dropping Function and Infinite Buffer

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Andrzej Chydzinski ◽  
Marek Barczyk ◽  
Dominik Samociuk
Keyword(s):  
Queueing Systems ◽  
Performance Characteristics ◽  
System Stability ◽  
System Response ◽  
Single Server ◽  
Queue Size ◽  
Finite Buffers ◽  
Guarantee System ◽  
Multimedia Transmissions ◽  
Dropping Function

We present an analysis of queues with the dropping function and infinite buffer. In such queues, the arriving packet (job, customer, etc.) can be dropped with the probability which is a function of the queue size. Currently, the main application area of the dropping function is active queue management in routers, but it is applicable also in many other queueing systems. So far, queues with the dropping function have been analyzed with finite buffers only, which led to complicated, computationally demanding formulas. Assuming infinite buffers enabled us herein to obtain formulas in compact, easy to use forms. Moreover, a model with the infinite buffer can often be used as a good approximation of the real queue, in which the buffer is large. We start with noticing that the classic stability condition, ρ<1, cannot be used for queues with the dropping function and infinite buffer. For this reason, we prove a few new, easy to use conditions, which guarantee system stability or instability. Then we prove several theorems on popular performance characteristics, including the queue size, busy period, loss ratio, output rate, and system response time. Additionally, we derive a special, very important characteristic called the burst ratio, which may influence severely the quality of real-time multimedia transmissions. All the theorems are illustrated with numerical examples, demonstrating in particular how the system stability may be tested and how the shape of the dropping function may affect different performance characteristics.

ON TRUNCATION PROPERTIES OF FINITE-BUFFER QUEUES AND QUEUEING NETWORKS

2000 ◽  
Vol 14 (4) ◽  
pp. 409-423 ◽  
Author(s):  
Xiuli Chao ◽  
Masakiyo Miyazawa
Keyword(s):  
Stochastic Process ◽  
Continuous Time ◽  
Queueing Networks ◽  
Additional Condition ◽  
Queueing Systems ◽  
Batch Service ◽  
Single Server ◽  
Finite Buffer ◽  
Finite Buffers ◽  
Batch Service Queues

We show that several truncation properties of queueing systems are consequences of a simple property of censored stochastic processes. We first consider a discrete-time stochastic process and show that its censored process has a truncated stationary distribution. When the stochastic process has continuous time, we present a similar result under the additional condition that the process is locally balanced. We apply these results to single-server batch arrival batch service queues with finite buffers and queueing networks with finite buffers and batch movements, and extend the well-known results on truncation properties of the MX/G/1/k queues and queueing networks with jump-over blocking.

scholarly journals Isomorphs of a class of Queueing systems

1979 ◽  
Vol 2 (1) ◽  
pp. 103-110
Author(s):  
A. Ghosal ◽  
Sudhir Madan
Keyword(s):  
Distribution Function ◽  
Waiting Time ◽  
Queueing Systems ◽  
Idle Time ◽  
Single Server ◽  
Queue Size ◽  
Restricted Sense

Restricted isomorphism between two queueing systems implies that they have equivalent distribution function for at least one (but not all) output elements (e.g. waiting time, queue size, idle time, etc.). Quasi-isomorphism implies an approximate equivalence. Most of the single-server queueing systems can be approximated by a quasi-isomorphic system which has a gamma inter-arrival and gamma service distributions(Ep/Eq/1).This paper deals with the derivation of simpler isomorphs (in quasi-restricted sense) for such gamma-gamma(Ep/Eq/1)systems.

scholarly journals MMAP/(PH,PH)/1 Queue with Priority Loss through Feedback

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1797
Author(s):  
Divya Velayudhan Nair ◽  
Achyutha Krishnamoorthy ◽  
Agassi Melikov ◽  
Sevinj Aliyeva
Keyword(s):  
Steady State ◽  
Numerical Experiments ◽  
Queueing Systems ◽  
Priority Queue ◽  
Performance Characteristics ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Phase Type ◽  
Phase Type Distributions

In this paper, we consider two single server queueing systems to which customers of two distinct priorities (P1 and P2) arrive according to a Marked Markovian arrival process (MMAP). They are served according to two distinct phase type distributions. The probability of a P1 customer to feedback is θ on completion of his service. The feedback (P1) customers, as well as P2 customers, join the low priority queue. Low priority (P2) customers are taken for service from the head of the line whenever the P1 queue is found to be empty at the service completion epoch. We assume a finite waiting space for P1 customers and infinite waiting space for P2 customers. Two models are discussed in this paper. In model I, we assume that the service of P2 customers is according to a non-preemptive service discipline and in model II, the P2 customers service follow a preemptive policy. No feedback is permitted to customers in the P2 line. In the steady state these two models are compared through numerical experiments which reveal their respective performance characteristics.

Optimality of threshold policies in single-server queueing systems with server vacations

1991 ◽  
Vol 23 (02) ◽  
pp. 388-405 ◽  
Author(s):  
Awi Federgruen ◽  
Kut C. So
Keyword(s):  
Queueing Systems ◽  
Operating Cost ◽  
Rate Function ◽  
Batch Size ◽  
Critical Threshold ◽  
Single Server ◽  
Queue Size ◽  
Cost Rate ◽  
Holding Cost ◽  
Poisson Arrivals

In this paper we consider a class of single-server queueing systems with compound Poisson arrivals, in which, at service completion epochs, the server has the option of taking off for one or several vacations of random length. The cost structure consists of a holding cost rate specified by a general non-decreasing function of the queue size, fixed costs for initiating and terminating service, and a variable operating cost incurred for each unit of time that the system is in operation. We show under some weak conditions with respect to the holding cost rate function and the service time, vacation time and arrival batch size distributions that it is either optimal among all feasible (stationary and non-stationary) policies never to take a vacation, or it is optimal to take a vacation when the system empties out and to resume work when, upon completion of a vacation, the queue size is equal to or in excess of a critical threshold. These optimality results are generalized for several variants of this model.

Optimality of threshold policies in single-server queueing systems with server vacations

1991 ◽  
Vol 23 (2) ◽  
pp. 388-405 ◽  
Author(s):  
Awi Federgruen ◽  
Kut C. So
Keyword(s):  
Queueing Systems ◽  
Operating Cost ◽  
Rate Function ◽  
Batch Size ◽  
Critical Threshold ◽  
Single Server ◽  
Queue Size ◽  
Cost Rate ◽  
Holding Cost ◽  
Poisson Arrivals

In this paper we consider a class of single-server queueing systems with compound Poisson arrivals, in which, at service completion epochs, the server has the option of taking off for one or several vacations of random length. The cost structure consists of a holding cost rate specified by a general non-decreasing function of the queue size, fixed costs for initiating and terminating service, and a variable operating cost incurred for each unit of time that the system is in operation. We show under some weak conditions with respect to the holding cost rate function and the service time, vacation time and arrival batch size distributions that it is either optimal among all feasible (stationary and non-stationary) policies never to take a vacation, or it is optimal to take a vacation when the system empties out and to resume work when, upon completion of a vacation, the queue size is equal to or in excess of a critical threshold. These optimality results are generalized for several variants of this model.

scholarly journals A Taylor Series Approach for Service-Coupled Queueing Systems with Intermediate Load

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Ekaterina Evdokimova ◽  
Sabine Wittevrongel ◽  
Dieter Fiems
Keyword(s):  
Performance Measures ◽  
Queueing Systems ◽  
Poisson Processes ◽  
Series Expansions ◽  
Service Rate ◽  
Single Server ◽  
Queueing Model ◽  
State Space Explosion ◽  
Finite Queues ◽  
Service Rates

This paper investigates the performance of a queueing model with multiple finite queues and a single server. Departures from the queues are synchronised or coupled which means that a service completion leads to a departure in every queue and that service is temporarily interrupted whenever any of the queues is empty. We focus on the numerical analysis of this queueing model in a Markovian setting: the arrivals in the different queues constitute Poisson processes and the service times are exponentially distributed. Taking into account the state space explosion problem associated with multidimensional Markov processes, we calculate the terms in the series expansion in the service rate of the stationary distribution of the Markov chain as well as various performance measures when the system is (i) overloaded and (ii) under intermediate load. Our numerical results reveal that, by calculating the series expansions of performance measures around a few service rates, we get accurate estimates of various performance measures once the load is above 40% to 50%.

Stability of Ring-Type MEMS Gyroscopes Subjected to Stochastic Angular Speed Fluctuation

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Samuel F. Asokanthan ◽  
Soroush Arghavan ◽  
Mohamed Bognash
Keyword(s):  
Angular Velocity ◽  
Degrees Of Freedom ◽  
Microelectromechanical Systems ◽  
Damping Ratio ◽  
Operating Conditions ◽  
System Stability ◽  
System Response ◽  
Ring Type ◽  
Mems Gyroscopes ◽  
The Stability

Effect of stochastic fluctuations in angular velocity on the stability of two degrees-of-freedom ring-type microelectromechanical systems (MEMS) gyroscopes is investigated. The governing stochastic differential equations (SDEs) are discretized using the higher-order Milstein scheme in order to numerically predict the system response assuming the fluctuations to be white noise. Simulations via Euler scheme as well as a measure of largest Lyapunov exponents (LLEs) are employed for validation purposes due to lack of similar analytical or experimental data. The response of the gyroscope under different noise fluctuation magnitudes has been computed to ascertain the stability behavior of the system. External noise that affect the gyroscope dynamic behavior typically results from environment factors and the nature of the system operation can be exerted on the system at any frequency range depending on the source. Hence, a parametric study is performed to assess the noise intensity stability threshold for a number of damping ratio values. The stability investigation predicts the form of threshold fluctuation intensity dependence on damping ratio. Under typical gyroscope operating conditions, nominal input angular velocity magnitude and mass mismatch appear to have minimal influence on system stability.

The single-server queue with service depending on queue size and with the preemptive-resume last-come–first-served queue discipline

1987 ◽  
Vol 24 (03) ◽  
pp. 758-767
Author(s):  
D. Fakinos
Keyword(s):  
Queueing System ◽  
Limiting Distribution ◽  
Single Server ◽  
Queue Size ◽  
Single Server Queue ◽  
Preemptive Resume ◽  
Server Queue ◽  
Queue Discipline ◽  
Service Times

This paper studies theGI/G/1 queueing system assuming that customers have service times depending on the queue size and also that they are served in accordance with the preemptive-resume last-come–first-served queue discipline. Expressions are given for the limiting distribution of the queue size and the remaining durations of the corresponding services, when the system is considered at arrival epochs, at departure epochs and continuously in time. Also these results are applied to some particular cases of the above queueing system.

Relations Among Moments of Waiting Time and Queue Size Distribution in Bulk Queueing Systems

1987 ◽  
Vol 36 (1-2) ◽  
pp. 63-68
Author(s):  
A. Ghosal ◽  
S. Madan ◽  
M.L. Chaudhry
Keyword(s):  
Size Distribution ◽  
Waiting Time ◽  
Queueing Systems ◽  
Queueing Models ◽  
Queue Size ◽  
Different Types

This paper brings out relations among the moments of various orders of the waiting time and the queue size in different types of bulk queueing models.

On queueing systems by retrials

1983 ◽  
Vol 20 (02) ◽  
pp. 380-389 ◽  
Author(s):  
Vidyadhar G. Kulkarni
Keyword(s):  
General Result ◽  
Service Time ◽  
Queueing Systems ◽  
Single Server ◽  
Expected Number ◽  
Waiting Room ◽  
Second Moments ◽  
Different Types ◽  
General Distributions ◽  
Arbitrary Service Time

A general result for queueing systems with retrials is presented. This result relates the expected total number of retrials conducted by an arbitrary customer to the expected total number of retrials that take place during an arbitrary service time. This result is used in the analysis of a special system where two types of customer arrive in an independent Poisson fashion at a single-server service station with no waiting room. The service times of the two types of customer have independent general distributions with finite second moments. When the incoming customer finds the server busy he immediately leaves and tries his luck again after an exponential amount of time. The retrial rates are different for different types of customers. Expressions are derived for the expected number of retrial customers of each type.

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