scholarly journals Two parallel finite queues with simultaneous services and Markovian arrivals

1997 ◽  
Vol 10 (4) ◽  
pp. 383-405 ◽  
Author(s):  
S. R. Chakravarthy ◽  
S. Thiagarajan
Keyword(s):  
Markov Process ◽  
Performance Measures ◽  
System Performance ◽  
Arrival Process ◽  
Single Server ◽  
Queueing Model ◽  
Finite Capacity ◽  
Numerical Examples ◽  
Finite Queues ◽  
Large State Space

In this paper, we consider a finite capacity single server queueing model with two buffers, A and B, of sizes K and N respectively. Messages arrive one at a time according to a Markovian arrival process. Messages that arrive at buffer A are of a different type from the messages that arrive at buffer B. Messages are processed according to the following rules: 1. When buffer A(B) has a message and buffer B(A) is empty, then one message from A(B) is processed by the server. 2. When both buffers, A and B, have messages, then two messages, one from A and one from B, are processed simultaneously by the server. The service times are assumed to be exponentially distributed with parameters that may depend on the type of service. This queueing model is studied as a Markov process with a large state space and efficient algorithmic procedures for computing various system performance measures are given. Some numerical examples are discussed.

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%.

A MULTI-SERVER QUEUEING MODEL WITH MARKOVIAN ARRIVALS AND MULTIPLE THRESHOLDS

2007 ◽  
Vol 24 (02) ◽  
pp. 223-243 ◽  
Author(s):  
SRINIVAS R. CHAKRAVARTHY
Keyword(s):  
Optimization Problem ◽  
Arrival Process ◽  
Single Server ◽  
Queueing Model ◽  
Process Map ◽  
Practical Applications ◽  
Setup Costs ◽  
Multiple Thresholds ◽  
Minimum Delay ◽  
Multi Server

We consider a multi-server queueing model in which arrivals occur according to a Markovian arrival process (MAP). There is a single-server and additional (backup) servers are added or removed depending on sets of thresholds. The service times are assumed to be exponential and the servers are assumed to be homogeneous. A comparison of this model to the classical MAP/M/c queueing model through an optimization problem yields some interesting results that are useful in practical applications. For example, we notice that positively correlated arrival process appears to benefit with the threshold type queueing model. We also give the minimum delay costs and the associated maximum setup costs so that the threshold type queueing model is to be preferred over the classical MAP/M/c model.

scholarly journals Some performance measures for vacation models with a batch Markovian arrival process

1994 ◽  
Vol 7 (2) ◽  
pp. 111-124 ◽  
Author(s):  
Sadrac K. Matendo
Keyword(s):  
Performance Measures ◽  
Waiting Times ◽  
Distribution Functions ◽  
Markovian Arrival Process ◽  
Arrival Process ◽  
Service Discipline ◽  
Renewal Processes ◽  
Single Server ◽  
Batch Markovian Arrival Process ◽  
Interarrival Times

We consider a single server infinite capacity queueing system, where the arrival process is a batch Markovian arrival process (BMAP). Particular BMAPs are the batch Poisson arrival process, the Markovian arrival process (MAP), many batch arrival processes with correlated interarrival times and batch sizes, and superpositions of these processes. We note that the MAP includes phase-type (PH) renewal processes and non-renewal processes such as the Markov modulated Poisson process (MMPP).The server applies Kella's vacation scheme, i.e., a vacation policy where the decision of whether to take a new vacation or not, when the system is empty, depends on the number of vacations already taken in the current inactive phase. This exhaustive service discipline includes the single vacation T-policy, T(SV), and the multiple vacation T-policy, T(MV). The service times are i.i.d. random variables, independent of the interarrival times and the vacation durations. Some important performance measures such as the distribution functions and means of the virtual and the actual waiting times are given. Finally, a numerical example is presented.

scholarly journals Maximum Entropy Approach for Un-Reliable Server Vacation Queueing Model with Optional Bulk Service

2012 ◽  
Vol 23 (1) ◽  
pp. 89-113
Author(s):  
Madhu Jain, Madhu Jain,
Keyword(s):  
Maximum Entropy ◽  
Performance Measures ◽  
Probability Generating Function ◽  
Single Server ◽  
Queueing Model ◽  
Time Service ◽  
Server Vacation ◽  
Single Vacation ◽  
Bulk Service ◽  
Essential Service

In this study, we consider a single server vacation queueing model with optional bulk service and an un-reliable server. A single server provides first essential service (FES) to all arriving customers one by one; apart from essential service, he can also facilitate the additional phase of optional service (OS) in batches of fixed size b( ≥ 1), in case when the customers request for it. The server may take a single vacation whenever he finds no customers waiting in the queue to be served. Moreover, the server is subjected to unpredictable breakdown while providing the first essential service. The vacation time, service time and repair time of the server are exponentially distributed. The steady state results are obtained in terms of probability generating function for queue size distributions. By using the maximum entropy analysis (MEA), we derive various system performance measures. A comparative study is performed between the exact and approximate waiting time of the system. By taking the numerical illustrations, the sensitivity analysis is done to explore the effect of different descriptors on various performance measures.

Analysis of finite-capacity polling systems

1991 ◽  
Vol 23 (2) ◽  
pp. 373-387 ◽  
Author(s):  
Hideaki Takagi
Keyword(s):  
Time Distribution ◽  
Arrival Process ◽  
Polling Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
Finite Capacity ◽  
Poisson Arrival ◽  
Finite Capacity Queues ◽  
General Service ◽  
Limited Service

We consider a system of N finite-capacity queues attended by a single server in cyclic order. For each visit by the server to a queue, the service is given continuously until that queue becomes empty (exhaustive service), given continuously only to those customers present at the visiting instant (gated service), or given to only a single customer (limited service). The server then switches to the next queue with a random switchover time, and administers the same type of service there similarly. For such a system where each queue has a Poisson arrival process, general service time distribution, and finite capacity, we find the distribution of the waiting time at each queue by utilizing the known results for a single M/G/1/K queue with multiple vacations.

scholarly journals Geo/Geo/1/N Queue with In-Immune and Immune Service Killing Discipline

2012 ◽  
Vol 23 (1) ◽  
pp. 129-148
Author(s):  
Madhu Jain Madhu Jain
Keyword(s):  
Performance Measures ◽  
Generating Functions ◽  
Relaxation Method ◽  
Single Server ◽  
Finite Capacity ◽  
Single Server Queue ◽  
Stationary Probability Distribution ◽  
Negative Customers ◽  
Server Queue ◽  
Successive Over Relaxation

The present investigation studies a discrete time single server queue with both positive and negative arrival streams in accordance with removal of the customer from the end (RCE)-in immune and immune service killing policy. This study is a generalization of the queue with negative customers, wherein only positive customers need a service and negative customers arriving to the system can kill the already present positive customers from any where in the queue, otherwise get lost. The concept of both in-immune and immune service killing are taken into consideration. According to the in-immune killing policy, the negative customer is allowed to kill the most recent positive customer inspite of whether it is in service or not, while the immune service killing discipline suggests that the customer currently being served is immune from killing by the negative arrival. We analyze a queue with geometric arrivals of both positive and negative customers for a finite capacity system. The stationary probability distribution and other performance measures are derived in terms of the generating functions. The results so obtained are validated by the numerical method based on successive over relaxation method (SOR). We have also employed the neurro fuzzy approach for exhibiting the approximate results for various performance measures.

scholarly journals Performance Measures of State Dependent MMPP/M/1 Queue

2018 ◽  
Vol 7 (4.10) ◽  
pp. 942 ◽  
Author(s):  
R. Sakthi ◽  
V. Vidhya ◽  
K. Mahaboob Hassain Sherieff ◽  
. .
Keyword(s):  
Performance Measures ◽  
Research Work ◽  
Arrival Process ◽  
Service Rate ◽  
Single Server ◽  
State Dependent ◽  
Markov Modulated ◽  
Service State ◽  
Modulated Process ◽  
Stationary Probability Vector

In this research work we are concerned with single unit server queue  queue with Markov Modulated process in Poisson fashion and the service time follow exponential distribution. The system is framed as a state dependent with the arrival process as Markov Modulated input and service is rendered by a single server with variation in service rate based on the intensity of service state of the system. The rate matrix that is essential to compute the stationary probability vector is obtained and various performance measures are computed using matrix method.

Analysis of finite-capacity polling systems

1991 ◽  
Vol 23 (02) ◽  
pp. 373-387 ◽  
Author(s):  
Hideaki Takagi
Keyword(s):  
Time Distribution ◽  
Arrival Process ◽  
Polling Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
Finite Capacity ◽  
Poisson Arrival ◽  
Finite Capacity Queues ◽  
General Service ◽  
Limited Service

We consider a system of N finite-capacity queues attended by a single server in cyclic order. For each visit by the server to a queue, the service is given continuously until that queue becomes empty (exhaustive service), given continuously only to those customers present at the visiting instant (gated service), or given to only a single customer (limited service). The server then switches to the next queue with a random switchover time, and administers the same type of service there similarly. For such a system where each queue has a Poisson arrival process, general service time distribution, and finite capacity, we find the distribution of the waiting time at each queue by utilizing the known results for a single M/G/1/K queue with multiple vacations.

A FINITE CAPACITY RESEQUENCING MODEL WITH MARKOVIAN ARRIVALS

2005 ◽  
Vol 22 (03) ◽  
pp. 409-443 ◽  
Author(s):  
SRINIVAS R. CHAKRAVARTHY ◽  
STEFANKA CHUKOVA
Keyword(s):  
Steady State ◽  
Optimization Problems ◽  
Arrival Process ◽  
Finite Buffer ◽  
Finite Capacity ◽  
Departure Process ◽  
Queuing Model ◽  
Process Map ◽  
Steady State Analysis ◽  
Numerical Examples

In this paper, we consider a two-server finite capacity queuing model in which messages should leave the system in the order in which they entered the system. Messages arrive according to a Markovian arrival process (MAP) and any message finding the buffer full is considered lost. Out-of-sequence messages are stored in a (finite) buffer and may lead to blocking when a processed message cannot be placed in the buffer. The steady state analysis of the model is performed by exploiting the structure of the coefficient matrices. The departure process is characterized and two interesting optimization problems along with illustrative numerical examples are discussed.

scholarly journals A finite capacity bulk service queue with single vacation and Markovian arrival process

2004 ◽  
Vol 2004 (4) ◽  
pp. 337-357 ◽  
Author(s):  
U. C. Gupta ◽  
Karabi Sikdar
Keyword(s):  
Asynchronous Transfer Mode ◽  
Maximum Size ◽  
Markovian Arrival Process ◽  
Telecommunication Networks ◽  
Arrival Process ◽  
Single Server ◽  
Finite Capacity ◽  
Single Vacation ◽  
Service Completion ◽  
And Performance

Vacation time queues with Markovian arrival process (MAP) are mainly useful in modeling and performance analysis of telecommunication networks based on asynchronous transfer mode (ATM) environment. This paper analyzes a single-server finite capacity queue wherein service is performed in batches of maximum size “b” with a minimum threshold “a” and arrivals are governed by MAP. The server takes a single vacation when he finds less than “a” customers after service completion. The distributions of buffer contents at various epochs (service completion, vacation termination, departure, arbitrary and pre-arrival) have been obtained. Finally, some performance measures such as loss probability and average queue length are discussed. Numerical results are also presented in some cases.

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