Optimal Server Allocation to Parallel Queues with Finite-Capacity Buffers

1996 ◽  
Vol 10 (2) ◽  
pp. 279-285 ◽  
Author(s):  
K. M. Wasserman ◽  
Nicholas Bambos
Keyword(s):  
Arbitrary Distribution ◽  
Single Server ◽  
Finite Capacity ◽  
Dynamic Allocation ◽  
Allocation Policy ◽  
Parallel Queues ◽  
Server Allocation ◽  
Arrival Processes ◽  
Service Times ◽  
Number Of Customers

In this paper, we study the problem of dynamic allocation of a single server to parallel queues with finite-capacity buffers. The arrival processes are mutually independent, equal rate Poisson processes, and the service times are independent and identically distributed random variables with an arbitrary distribution. We are interested in characterizing the allocation policy that stochastically minimizes the number of customers lost due to buffer overflows. Using a coupling argument, we establish the optimality of the Fewest-Empty-Spaces policy, which allocates the server to the queue with the fewest empty buffer spaces, within the class of nonpreemptive and nonidling policies. The result extends to the class of preemptive policies, if the service times are exponentially distributed. We also briefly discuss the allocation problem under more general statistical assumptions on the arrival processes.

OPTIMAL CONTROL OF PARALLEL QUEUES WITH BATCH SERVICE

2002 ◽  
Vol 16 (3) ◽  
pp. 289-307 ◽  
Author(s):  
Cathy H. Xia ◽  
George Michailidis ◽  
Nicholas Bambos ◽  
Peter W. Glynn
Keyword(s):  
Fluid Model ◽  
Heavy Traffic ◽  
Batch Size ◽  
Single Server ◽  
Dynamic Allocation ◽  
Finite Buffers ◽  
Threshold Policy ◽  
Parallel Queues ◽  
Buffer Overflows ◽  
Arrival Processes

We consider the problem of dynamic allocation of a single server with batch processing capability to a set of parallel queues. Jobs from different classes cannot be processed together in the same batch. The arrival processes are mutually independent Poisson flows with equal rates. Batches have independent and identically distributed exponentially distributed service times, independent of the batch size and the arrival processes. It is shown that for the case of infinite buffers, allocating the server to the longest queue, stochastically maximizes the aggregate throughput of the system. For the case of equal-size finite buffers the same policy stochastically minimizes the loss of jobs due to buffer overflows. Finally, for the case of unequal-size buffers, a threshold-type policy is identified through an extensive simulation study and shown to consistently outperform other conventional policies. The good performance of the proposed threshold policy is confirmed in the heavy-traffic regime using a fluid model.

Extremal properties of the FIFO discipline in queueing networks

1992 ◽  
Vol 29 (4) ◽  
pp. 967-978 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar
Keyword(s):  
Likelihood Ratio ◽  
Service Time ◽  
Queueing Networks ◽  
Service Rate ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queues ◽  
First Results ◽  
Service Times ◽  
Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

A service model in which the server is required to search for customers

1984 ◽  
Vol 21 (1) ◽  
pp. 157-166 ◽  
Author(s):  
Marcel F. Neuts ◽  
M. F. Ramalhoto
Keyword(s):  
Poisson Process ◽  
Numerical Computation ◽  
Probability Distributions ◽  
General Distribution ◽  
Search Process ◽  
Single Server ◽  
Stationary Probability ◽  
Service Model ◽  
Service Times ◽  
Number Of Customers

Customers enter a pool according to a Poisson process and wait there to be found and processed by a single server. The service times of successive items are independent and have a common general distribution. Successive services are separated by seek phases during which the server searches for the next customer. The search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t. Various stationary probability distributions for this model are obtained in explicit forms well-suited for numerical computation.Under the assumption of exponential service times, corresponding results are obtained for the case where customers may escape from the pool.

A solvable model for a finite-capacity queueing system

1985 ◽  
Vol 22 (4) ◽  
pp. 903-911 ◽  
Author(s):  
V. Giorno ◽  
C. Negri ◽  
A. G. Nobile
Keyword(s):  
Queueing System ◽  
Waiting Times ◽  
Queueing Systems ◽  
Solvable Model ◽  
Probability Density Functions ◽  
Single Server ◽  
Finite Capacity ◽  
System Service ◽  
Transient Probabilities ◽  
Number Of Customers

Single–server–single-queue–FIFO-discipline queueing systems are considered in which at most a finite number of customers N can be present in the system. Service and arrival rates are taken to be dependent upon that state of the system. Interarrival intervals, service intervals, waiting times and busy periods are studied, and the results obtained are used to investigate the features of a special queueing model characterized by parameters (λ (Ν –n), μn). This model retains the qualitative features of the C-model proposed by Conolly [2] and Chan and Conolly [1]. However, quite unlike the latter, it also leads to closed-form expressions for the transient probabilities, the interarrival and service probability density functions and their moments, as well as the effective interarrival and service densities and their moments. Finally, some computational results are given to compare the model discussed in this paper with the C-model.

scholarly journals Analysis of Single-Server Queue with Phase-Type Service and Energy Harvesting

2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Sergey A. Dudin ◽  
Moon Ho Lee
Keyword(s):  
Sensor Node ◽  
Random Number ◽  
Single Server ◽  
Finite Capacity ◽  
Central Node ◽  
Phase Type ◽  
Server Queue ◽  
Arrival Processes ◽  
System States ◽  
Little's Formula

We propose a queueing model suitable, for example, for modelling operation of nodes of sensor networks. The sensor node senses a random field and generates packets to be transmitted to the central node. The sensor node has a battery of a finite capacity and harvests energy during its operation from outside (using solar cells, wind turbines, piezoelectric cells, etc.). We assume that, generally speaking, service (transmission) of a packet consists of a random number of phases and implementation of each phase requires a unit of energy. If the battery becomes empty, transmission is failed. To reduce the probability of forced transmission termination, we suggest that the packet can be accepted for transmission only when the number of energy units is greater than or equal to some threshold. Under quite general assumptions about the pattern of the arrival processes of packets and energy, we compute the stationary distributions of the system states and the waiting time of a packet in the system and numerically analyze performance measures of the system as functions of the threshold. Validity of Little’s formula and its counterpart is verified.

A solvable model for a finite-capacity queueing system

1985 ◽  
Vol 22 (04) ◽  
pp. 903-911 ◽  
Author(s):  
V. Giorno ◽  
C. Negri ◽  
A. G. Nobile
Keyword(s):  
Queueing System ◽  
Waiting Times ◽  
Queueing Systems ◽  
Solvable Model ◽  
Probability Density Functions ◽  
Single Server ◽  
Finite Capacity ◽  
System Service ◽  
Transient Probabilities ◽  
Number Of Customers

Single–server–single-queue–FIFO-discipline queueing systems are considered in which at most a finite number of customers N can be present in the system. Service and arrival rates are taken to be dependent upon that state of the system. Interarrival intervals, service intervals, waiting times and busy periods are studied, and the results obtained are used to investigate the features of a special queueing model characterized by parameters (λ (Ν –n), μn). This model retains the qualitative features of the C-model proposed by Conolly [2] and Chan and Conolly [1]. However, quite unlike the latter, it also leads to closed-form expressions for the transient probabilities, the interarrival and service probability density functions and their moments, as well as the effective interarrival and service densities and their moments. Finally, some computational results are given to compare the model discussed in this paper with the C-model.

A service model in which the server is required to search for customers

1984 ◽  
Vol 21 (01) ◽  
pp. 157-166 ◽  
Author(s):  
Marcel F. Neuts ◽  
M. F. Ramalhoto
Keyword(s):  
Poisson Process ◽  
Numerical Computation ◽  
Probability Distributions ◽  
General Distribution ◽  
Search Process ◽  
Single Server ◽  
Stationary Probability ◽  
Service Model ◽  
Service Times ◽  
Number Of Customers

Customers enter a pool according to a Poisson process and wait there to be found and processed by a single server. The service times of successive items are independent and have a common general distribution. Successive services are separated by seek phases during which the server searches for the next customer. The search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t. Various stationary probability distributions for this model are obtained in explicit forms well-suited for numerical computation. Under the assumption of exponential service times, corresponding results are obtained for the case where customers may escape from the pool.

Extremal properties of the FIFO discipline in queueing networks

1992 ◽  
Vol 29 (04) ◽  
pp. 967-978 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar
Keyword(s):  
Likelihood Ratio ◽  
Service Time ◽  
Queueing Networks ◽  
Service Rate ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queues ◽  
First Results ◽  
Service Times ◽  
Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons: (i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means. (ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means. We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR. Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

scholarly journals Output process of the M|GI|1 is an asymptotical renewal process

2021 ◽  
Vol 21 (1) ◽  
pp. 100-110
Author(s):  
Ivan L. Lapatin ◽  
◽  
Anatoly A. Nazarov ◽  
Keyword(s):  
Distribution Function ◽  
Service Time ◽  
Renewal Process ◽  
Retrial Queue ◽  
Arbitrary Distribution ◽  
Single Server ◽  
Output Process ◽  
Limit Condition ◽  
Repeated Calls ◽  
Number Of Customers

Most of the studies on models with retrials are devoted to the research of the number of applications in the system or in the source of repeated calls using asymptotic and numerical approaches or simulation. Although one of the main characteristics that determines the quality of the communication system is the number of applications served by the system per unit of time. Information on the characteristics of the output processes is of great practical interest, since the output process of one system may be incoming to another. The results of the study of the outgoing flows of queuing networks are widely used in the modeling of computer systems, in the design of data transmission networks and in the analysis of complex multi-stage production processes. In this paper, we have considered a single server system with redial, the input of which receives a stationary Poisson process. The service time in considered system is a random value with an arbitrary distribution function B(x). If the customer enters the system and finds the server busy, it instantly joins the orbit and carries out a random delay there during an exponentially distributed time. The object of study is the output process of this system. The output is characterized by the probability distribution of the number of customers that have completed service for time t. We have provided the study using asymptotic analysis method under low rate of retrials limit condition. We have shown in the paper that the output of retrial queue M|GI|1 is an asymptotical renewal process. Moreover, the lengths of the intervals in output process are the sum of an exponential random value with the parameter lambda + kappa and a random variable with the distribution function B(x). The results of a numerical experiment show that the probability distributions of the number of served customers in the system are practically the same for significantly different distribution laws B(x) of service time if the service times have the same first two moments.

On a tandem queueing model with identical service times at both counters, I

1979 ◽  
Vol 11 (3) ◽  
pp. 616-643 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Waiting Times ◽  
Sufficient Conditions ◽  
Complete Analysis ◽  
Single Server ◽  
Stationary Distributions ◽  
In Series ◽  
Queues In Series ◽  
Service Times ◽  
Necessary And Sufficient ◽  
Insight Into

This paper considers a queueing system consisting of two single-server queues in series, in which the service times of an arbitrary customer at both queues are identical. Customers arrive at the first queue according to a Poisson process.Of this model, which is of importance in modern network design, a rather complete analysis will be given. The results include necessary and sufficient conditions for stationarity of the tandem system, expressions for the joint stationary distributions of the actual waiting times at both queues and of the virtual waiting times at both queues, and explicit expressions (i.e., not in transform form) for the stationary distributions of the sojourn times and of the actual and virtual waiting times at the second queue.In Part II (pp. 644–659) these results will be used to obtain asymptotic and numerical results, which will provide more insight into the general phenomenon of tandem queueing with correlated service times at the consecutive queues.

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