First-come First-served Discipline Research Articles

Fork–join and redundancy systems with heavy-tailed job sizes

We investigate the tail asymptotics of the response time distribution for the cancel-on-start (c.o.s.) and cancel-on-completion (c.o.c.) variants of redundancy-d scheduling and the fork–join model with heavy-tailed job sizes. We present bounds, which only differ in the pre-factor, for the tail probability of the response time in the case of the first-come first-served discipline. For the c.o.s. variant, we restrict ourselves to redundancy-d scheduling, which is a special case of the fork–join model. In particular, for regularly varying job sizes with tail index-nu the tail index of the response time for the c.o.s. variant of redundancy-d equals -min {d_{mathrm {cap}}(nu -1),nu }, where d_{mathrm {cap}} = min {d,N-k}, N is the number of servers and k is the integer part of the load. This result indicates that for d_{mathrm {cap}} < frac{nu }{nu -1} the waiting time component is dominant, whereas for d_{mathrm {cap}} > frac{nu }{nu -1} the job size component is dominant. Thus, having d = lceil min {frac{nu }{nu -1},N-k} rceil replicas is sufficient to achieve the optimal asymptotic tail behavior of the response time. For the c.o.c. variant of the fork–join (n_{mathrm {F}},n_{mathrm {J}}) model, the tail index of the response time, under some assumptions on the load, equals 1-nu and 1-(n_{mathrm {F}}+1-n_{mathrm {J}})nu , for identical and i.i.d. replicas, respectively; here, the waiting time component is always dominant.

Ergodic diffusion control of multiclass multi-pool networks in the Halfin–Whitt regime

We consider Markovian multiclass multi-pool networks with heterogeneous server pools, each consisting of many statistically identical parallel servers, where the bipartite graph of customer classes and server pools forms a tree. Customers form their own queue and are served in the first-come first-served discipline, and can abandon while waiting in queue. Service rates are both class and pool dependent. The objective is to study the limiting diffusion control problems under the long run average (ergodic) cost criteria in the Halfin–Whitt regime. Two formulations of ergodic diffusion control problems are considered: (i) both queueing and idleness costs are minimized, and (ii) only the queueing cost is minimized while a constraint is imposed upon the idleness of all server pools. We develop a recursive leaf elimination algorithm that enables us to obtain an explicit representation of the drift for the controlled diffusions. Consequently, we show that for the limiting controlled diffusions, there always exists a stationary Markov control under which the diffusion process is geometrically ergodic. The framework developed in [Ann. Appl. Probab. 25 (2015) 3511–3570] is extended to address a broad class of ergodic diffusion control problems with constraints. We show that the unconstrained and constrained problems are well posed, and we characterize the optimal stationary Markov controls via HJB equations.

Characteristics of rainfall queues for rain attenuation studies over radio links at subtropical and equatorial Africa

Attenuation due to precipitation remains an important design factor in the future deployment of terrestrial and earth-space communication radio links. Largely, there are concerted efforts to understand the dynamics of precipitation in attenuation occurrence at subtropical, tropical, and equatorial region of Africa. In this deliberate approach, rainfall spikes pertaining to rain cells are conceptualized as distinct rain spike traffic over radio links, by applying queueing theory concepts. The queue distributions at Durban (29°52′S, 30°58′E) and Butare (2°36′S, 29°44′E)—respectively, of subtropical and equatorial climates—are investigated from distrometer measurements. The data sets at both sites are observed over four rain regimes: drizzle, widespread, shower, and thunderstorm. The queue parameters of service time and inter-arrival of rain spikes traffic at both regions are found to be Erlang-k distributed (Ek) and exponentially distributed (M), respectively. It is established that the appearance of rain rates over radio links invariably follows a First Come, First Served (FCFS), multi-server (s), infinite queue, and semi-Markovian process, designated as M/Ek/s/∞/FCFS discipline. Modeled queue parameters at both regions are found to vary significantly over different regimes. However, these queue parameters over the entire data set suggest similar queue patterns at both sites. More importantly, power law relationships describing other queue-related parameters are formulated. The paper concludes by demonstrating an application of queueing theory for rainfall synthesis. The proposed technique will provide an alternative method of estimating rain cell sizes and rain attenuation over satellite and terrestrial links.

Performance analysis of call centers with abandonment, retrial and after-call work

This paper considers a multiserver queueing model with abandonment, retrial and after-call work for call centers. Upon a phone call, customers that find a free call line occupy the line immediately while those who see all the call lines busy are blocked and join an orbit. Customers holding a call line are served according to the first-come first-served discipline. After completing a call, the customer leaves the system while the server must start an after-call work and the call line is released for a newly arrived customer. Waiting customers may abandon after some waiting time and then either join the orbit or leave forever. Customers in the orbit retry to hold a free call line after some time. We formulate the queueing system using a continuous-time level-dependent quasi-birth-and-death process for which a sufficient condition for the ergodicity is derived. We obtain a numerical solution for the stationary distribution based on which performance measures such as the waiting time distribution and the blocking probability are derived. Using Little’s law, we obtain explicit formulae which verify the accuracy of the numerical solution. We compare our model with some simpler models which do not fully take into account some human behaviors. The comparison shows significant differences implying the importance of our model. Numerical results show various insights into the performance of call centers.

Workload bounds in fluid models with priorities

In this paper we establish upper and lower bounds on the steady-state per-class workload distributions in a single-server queue with multiple priority classes. Motivated by communication network applications, the model has constant processing rate and general input processes with stationary increments. The bounds involve corresponding quantities in related models with the first-come first-served discipline. We apply the bounds to support a new notion of effective bandwidths for multi-class systems with priorities. We also apply the lower bound to obtain sufficient conditions for the workload distributions to have heavy tails.

The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution

By exploiting an infinite-server-model lower bound, we show that the tails of the steady-state and transient waiting-time distributions in the M/GI/s queue with unlimited waiting room and the first-come first-served discipline are bounded below by tails of Poisson distributions. As a consequence, the tail of the steady-state waiting-time distribution is bounded below by a constant times the sth power of the tail of the service-time stationary-excess distribution. We apply that bound to show that the steady-state waiting-time distribution has a heavy tail (with appropriate definition) whenever the service-time distribution does. We also establish additional results that enable us to nearly capture the full asymptotics in both light and heavy traffic. The difference between the asymptotic behavior in these two regions shows that the actual asymptotic form must be quite complicated.

Performance analysis of broadcast star network with priorities

The authors present a model for the broadcast star network with collision avoidance switch and with multiple priority classes of messages. They also propose a service discipline which is easy to implement and exhibits relatively low variance of message delay. This service discipline, which is used to select a message among messages with the same class-index, is a combination of first-come first-served (FCFS) and random-order of service (ROS) disciplines, and it uses the priority fields in the message header. Then, the ready stations can be considered to form a distributed queue, and the entire system can be considered to form a priority M/G/1 queueing system with multiple server vacations. They analyze the model to obtain the distributions of the message delay for the proposed service discipline, and compare the results with that for the FCFS discipline. From numerical results, they show that the variance of the message delay for the proposed service discipline is almost the same as that for the FCFS discipline for any distribution of message length and also for any value of normalized propagation delay.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Performance analysis of broadcast star network with collision-avoidance switch

It is desirable to have a protocol which has low delay in light traffic condition and does not suffer from losing the channel capacity in heavy traffic condition. One solution to this problem is the use of a collision-avoidance switch. In this paper, we present models for broadcast star networks with collision-avoidance switch in two cases: a broadcast star network with small propagation delay and a broadcast star network with large propagation delay. Also, we propose a service discipline which is easy to implement and exhibits relatively low variance of message delay. The proposed service discipline is a combination of first-come first-served (FCFS) and random-order of service (ROS) discipline, and it uses reservation and priority fields in the message header. We analyze the models to obtain distributions of message delay for the proposed service discipline, and compare the results with that for other service disciplines. From numerical results, we show that the variance of message delay for the proposed service discipline is almost the same as that for the FCFS discipline for any distribution of message length and also for any value of normalized propagation delay.

Two-server queue with one server idle below a threshold

A birth-death queueing system with two identical servers, first-come first-served discipline, and Poisson arrivals is considered. Only one of the servers is active when the number of customers in the system does not exceed a prescribed threshold, whereas both are active above the threshold. The problem of determining the equilibrium density of the waiting time is formulated. A generating function is given for the Laplace transform of the density of the waiting time, and it is pointed out that it leads to an explicit expression for this quantity. Explicit expressions are obtained for the first and second moments of the waiting and sojourn times, and they are compared with the corresponding quantities for a single-server system with the same state-dependent mean service rates.

Limits for queues as the waiting room grows

We study the convergence of finite-capacity open queueing systems to their infinite-capacity counterparts as the capacity increases. Convergence of the transient behavior is easily established in great generality provided that the finite-capacity system can be identified with the infinite-capacity system up to the first time that the capacity is exceeded. Convergence of steady-state distributions is more difficult; it is established here for the GI/GI/c/n model withc servers,n-c extra waiting spaces and the first-come first-served discipline, in which all arrivals finding the waiting room full are lost without affecting future arrivals, via stochastic dominance and regenerative structure.

Single server with linear state-dependent mean rate

A birth-death queueing system with asingle server, first-come first-served discipline, Poisson arrivals and mean service rate which depends linearly on the number of customers in the system, is considered. Explicit expressions are derived for the equilibrium densities of the sojourn and waiting times. Simple approximations to the densities, including the first order correction terms, are obtained in a heavy-traffic situation.

Sojourn and waiting times in a single-server system with state-dependent mean service rate

A birth-death queueing system with asingle server, first-come first-served discipline, Poisson arrivals and state-dependent mean service rate is considered. The problem of determining the equilibrium densities of the sojourn and waiting times is formulated, in general. The particular case in which the mean service rate has one of two values, depending on whether or not the number of customers in the system exceeds a prescribed threshold, is then investigated. A generating function is derived for the Laplace transforms of the densities of the sojourn and waiting times, leading to explicit expressions for these quantities. Explicit expressions for the second moments of the sojourn and waiting times are also obtained.

Approximating the tandem behaviour of a G|M|1 queueing system

A heuristic algorithm is suggested for approximating the departure process from a G/M/1 queueing system with generally distributed interarrival times, a single markovian server, an infinite capacity, and the first-come first-served queueing discipline, in the steady state. The tandem behaviour of the system is then approximated in the steady state. Finally, the approximation results are compared against both a simulation study and the approximation results of Kuehn (1979) and Whitt (1983 a, b).

Certain optimality properties of the first-come first-served discipline for G/G/s queues

It is shown that for many server queueing systems in which arrivals are allocated to servers in a manner that does not depend on their service times, the vector of work-loads or virtual waiting times is least in the sense of weak majorization when the discipline is first-come first-served. Thus, the maximum work-load and the total work-load are minimized under this “intuitively obvious” minimizing discipline.

The influence of service-time variability in a closed network of queues

This paper describes the effect of service-time variability on the standard performance measures of a closed network of single-server queues with the first-come first-served discipline and one job class. Several service-time variability principles are proposed to serve as rough practical guidelines. The most interesting one states that the mean queue length at a bottleneck queue typically decreases when the variability of the service time at that queue is increased. The principles are supported here by numerical examples and theorems in special cases. The principles are also applied to test approximation procedures.

Departures from a Queue with Many Busy Servers

To analyze networks of queues, it is important to be able to analyze departure processes from single queues. For the M/M/s and M/G/∞ models, the stationary departure process is simple (Poisson), but in general the stationary departure process is quite complicated. As a basis for approximations, this paper shows that the stationary departure process is approximately Poisson when there are many busy slow servers in a large class of stationary G/GI/s congestion models having s servers, infinite waiting room, the first-come first-served discipline, and mutually independent and identically distributed service times that are independent of a stationary arrival process. Limit theorems are proved for the departure process in a G/GI/s system in which the number of servers and the offered load (arrival rate divided by the service rate) both increase. The asymptotic behavior of the departure process depends on the way the arrival rate changes. If the arrival rate is held fixed, so that the offered load increases by slowing down the service rate, then the departure process converges to a Poisson process. For this result, the service-time distribution is assumed to be phase-type. Other limiting behavior occurs if the arrival rate approaches zero or infinity. Convergence is established in each case by applying previous heavy-traffic limit theorems.

系GI/E_k/mにおける定常待ち分布

In this paper, the stationary waiting time distributions Fq (x) and F (x) are explicitly formulated for the GI/Ek/m queue under the first-come first-served discipline. The transition probability matrix and the imbedded probabilities play the important ro.les in this study. Some numerical results are presented for various systems as EQ/Ek/m, UQ/Ek/m, D/Ek/m, etc. Further the properties of F (x) are considered.

Performance Analysis of a Preemptive Priority Queue with Applications to Packet Communication Systems

In this paper we analyze the performance of a preemptive priority queue. We give the model description in the context of a packet communication system where message sources, having different priorities, share a common communication channel. Each source generates, as an independent Poisson process, messages consisting of an arbitrarily distributed, random number of fixed-length packets. The channel server can only begin service at integer multiples of the packet transmission time (i.e., a time-slotted channel is assumed), and the server will preempt an ongoing message transmission at the next packet boundary whenever there is a message arrival from a higher-priority source. The average in-queue waiting time for each packet in any given source message and the average message delay are derived along with the corresponding moment-generating functions. Also, comparisons are made with the first-come first-served queueing discipline.

The Machine Repair Problem with Heterogeneous Populations

The classic machine repair with spares (finite source) queueing model assumes all calling units are identical in failure and repair characteristics. This paper develops, for a first-come first-served discipline, a procedure for treating nonhomogeneous populations, specifically, a population with two types of items, each with exponential failure and repair times but with different mean values. Exact solutions are obtained for small population sizes and compared to approximate procedures using the classical theory. The exact model is essentially a two-stage cyclic queue with two classes of customers. Extensions of the model to N stages and M customer classes, priority disciplines, and other disciplines, including blocking, are discussed.

On multipopulation queuing systems with first‐come first‐served discipline

AbstractSteady‐state probabilities for a multipopulation single channel multiserver queuing system with first‐come first‐served queue discipline are established. Special cases lead to simplified formulas which sometimes can be evaluated using existing statistical and/or queuing tables. Optimization aspects are considered involving control of service rates. Several applications are presented.