Queues with Service Rate Controlled by a Delayed Feedback

We consider a single queue with a Markov modulated Poisson arrival process. Its service rate is controlled by a scheduler. The scheduler receives the workload information from the queue after a delay. This queue models the buffer in an earth station in a satellite network where the scheduler resides in the satellite. We obtain the conditions for stability, rates of convergence to the stationary distribution and the finiteness of the stationary moments. Next we extend these results to the system where the scheduler schedules the service rate among several competing queues based on delayed information about the workloads in the different queues.     

Expected Performance of a Queueing System with Ancillary Activities

Many service organizations follow a scheduling policy whereby employees alternate between direct customer service (e.g. bank deposits and withdrawals) and an ancillary activity (e.g. bookkeeping, paperwork). When queues are short, employees are diverted to ancillary activities; when queues become long, employees return to customer service. This paper evaluates trade-offs between the system objectives of (1) minimizing customer delay, (2) minimizing teller idle time, and (3) minimizing the disruption of ancillary activities, for queueing systems with ancillary activities. Customers are assumed to arrive by a stationary Poisson process, and service times are exponential random variables. In terms of objectives (1) and (3), the best of four policies studied is to add a server when the queue size per server reaches a maximum and remove a server when the queue size equals zero. The difference in teller idle time between the policies is generally small, particularly when the traffic intensity is large.     

A two-queue model with Bernoulli service schedule and switching times

In this paper, we present a detailed analysis of a cyclic-service queueing system consisting of two parallel queues, and a single server. The server serves the two queues with a Bernoulli service schedule described as follows. At the beginning of each visit to a queue, the server always serves a customer. At each epoch of service completion in the ith queue at which the queue is not empty, the server makes a random decision: with probability p_i, it serves the next customer; with probability 1-p_i, it switches to the other queue. The server takes switching times in its transition from one queue to the other. We derive the generating functions of the joint stationary queue-length distribution at service completion instants, by using the approach of the boundary value problem for complex variables. We also determine the Laplace-Stieltjes transforms of waiting time distributions for both queues, and obtain their mean waiting times. This revised version was published online in June 2006 with corrections to the Cover Date.     

Ergodicity and analysis of the process describing the system state in polling systems with two queues

Consider a polling system of two queues served by a single server that visits the queues in cyclic order. The polling discipline in each queue is of exhaustive-type, and zero-switchover times are considered. We assume that the arrival times in each queue form a Poisson process and that the service times form sequences of independent and identically distributed random variables, except for the service distribution of the first customer who is served at each polling instant (the time in which the server moves from one queue to the other one). The sufficient and necessary conditions for the ergodicity of such polling system are established as well as the stationary distribution for the continuous-time process describing the state of the system. The proofs rely on the combination of three embedded processes that were previously used in the literature. An important result is that ρ=1 can imply ergodicity in one specific case, where ρ is the typical traffic intensity for polling systems, and ρ<1 is the classical non-saturation condition.     

Interoutput times in processor sharing queues with feedback

While properties of the flows in isolated processor sharing queues are well understood, little is known about the flows in networks with processor sharing nodes. This paper analyzes the internal traffic processes in processor sharing queues with instantaneous Bernoulli feedback. The internal traffic does not inherit the insensitivity to the shape of the service requirement distribution from the external traffic. The interoutput time distribution is studied in the single server and infinite server processor sharing queues. For the systems we study, we show that when service requirement distributions with the same means are convexly ordered, so are interoutput time distributions.This work was partially supported by the National Science Foundation under Grant ECS-8501217 and by the Graduate School of the University of Massachusetts under Faculty Research Grant 1-03205.     

A tandem GI/PH/1→•/PH/1/0 queue with blocking

Tandem queues are widely used in mathematical modeling of random processes describing the operation of manufacturing systems, supply chains, computer and telecommunication networks. Although there exists a lot of publications on tandem queueing systems, analytical research on tandem queues with non-Markovian input is very limited. In this paper, the results of analytical investigation of two-node tandem queue with arbitrary distribution of inter-arrival times are presented. The first station of the tandem is represented by a single-server queue with infinite waiting room. After service at the first station, a customer proceeds to the second station that is described by a single-server queue without a buffer. Service times of a customer at the first and the second server have PH (Phase-type) distributions. A customer, who completes service at the first server and meets a busy second server, is forced to wait at the first server until the second server becomes available. During the waiting period, the first server becomes blocked, i.e., not available for service of customers. We calculate the joint stationary distribution of the system states at the embedded epochs and at arbitrary time. The Laplace–Stieltjes transform of the sojourn time distribution is derived. Key performance measures are calculated and numerical results presented.     

Symmetric queues served in cyclic order

Consider a symmetrical system of n queues served in cyclic order by a single server. It is shown that the stationary number of customers in the system is distributed as the sum of three independent random variables, one being the stationary number of customers in a standard M/G/1 queue. This fact is used to establish an upper bound for the mean waiting time for the case where at most k customers are served at each queue per visit by the server. This approach is also used to rederive the mean waiting times for the cases of exhaustive service, gated service, and serve at most one customer at each queue per visit by the server.     

A Communication Multiplexer Problem: Two Alternating Queues with Dependent Randomly-Timed Gated Regime

Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace–Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling instants and at an arbitrary moment, mean busy period duration and mean cycle time length, expected number of messages transmitted during a busy period and mean waiting times. Finally, we present graphs of numerical results comparing the mean waiting times in the two queues as functions of the relative loads, showing the effect of the RTG regime.     

Analysis of Alternating-Priority Queueing Models with (Cross) Correlated Switchover Times

This paper analyzes a single server queueing system in which service is alternated between two queues and the server requires a (finite) switchover time to switch from one queue to the other. The distinction from classical results is that the sequence of switchover times from each of the queues need not be i.i.d. nor independent from each other; each sequence is merely required to form a stationary ergodic sequence. With the help of stochastic recursive equations explicit expressions are derived for a number of performance measures, most notably for the average delay of a customer and the average queue lengths under different service disciplines. With these expressions a comparison is made between the service disciplines and the influence of correlation is studied. Finally, through a number of examples it is shown that the correlation can significantly increase the mean delay and the average queue lengths indicating that the correlation between switchover times should not be ignored. This has important implications for communication systems in which a common communication channel is shared amongst various users and where the time between consecutive data transfers is correlated (for example in ad-hoc networks). In addition to this a number of notational mistakes in well-known existing literature are pointed out. AMS subject classification: 68M20, 60J85 A shorter version of this work has been published in the proceedings of IEEE Infocom 2005. This work was partly sponsored by the EURONGI network of excellence.     

Some limit theorems for regenerative queues

We consider a single server queue with the interarrival times and the service times forming a regenerative sequence. This traffic class includes the standard models: iid, periodic, Markov modulated (e.g., BMAP model of Lucantoni [18]) and their superpositions. This class also includes the recently proposed traffic models in high speed networks, exhibiting long range dependence. Under minimal conditions we obtain the rates of convergence to stationary distributions, finiteness of stationary moments, various functional limit theorems and the continuity of stationary distributions and moments. We use the continuity results to obtain approximations for stationary distributions and moments of an MMPP/GI/1 queue where the modulating chain has a countable state space. We extend all our results to feed-forward networks where the external arrivals to each queue can be regenerative. In the end we show that the output process of a leaky bucket is regenerative if the input process is and hence our results extend to a queue with arrivals controlled by a leaky bucket. This revised version was published online in June 2006 with corrections to the Cover Date.     

Heavy-traffic limits for stationary network flows

This paper studies stationary customer flows in an open queueing network. The flows are the processes counting customers flowing from one queue to another or out of the network. We establish the existence of unique stationary flows in generalized Jackson networks and convergence to the stationary flows as time increases. We establish heavy-traffic limits for the stationary flows, allowing an arbitrary subset of the queues to be critically loaded. The heavy-traffic limit with a single bottleneck queue is especially tractable because it yields limit processes involving one-dimensional reflected Brownian motion. That limit plays an important role in our new nonparametric decomposition approximation of the steady-state performance using indices of dispersion and robust optimization.

     

Stability,monotonicity and invariant quantities in general polling systems

Consider a polling system withK1 queues and a single server that visits the queues in a cyclic order. The polling discipline in each queue is of general gated-type or exhaustive-type. We assume that in each queue the arrival times form a Poisson process, and that the service times, the walking times, as well as the set-up times form sequences of independent and identically distributed random variables. For such a system, we provide a sufficient condition under which the vector of queue lengths is stable. We treat several criteria for stability: the ergodicity of the process, the geometric ergodicity, and the geometric rate of convergence of the first moment. The ergodicity implies the weak convergence of station times, intervisit times and cycle times. Next, we show that the queue lengths, station times, intervisit times and cycle times are stochastically increasing in arrival rates, in service times, in walking times and in setup times. The stability conditions and the stochastic monotonicity results are extended to the polling systems with additional customer routing between the queues, as well as bulk and correlated arrivals. Finally, we prove that the mean cycle time, the mean intervisit time and the mean station times are invariant under general service disciplines and general stationary arrival and service processes.     

The cyclic queue and the tandem queue

We consider a closed queueing network, consisting of two FCFS single server queues in series: a queue with general service times and a queue with exponential service times. A fixed number \(N\) of customers cycle through this network. We determine the joint sojourn time distribution of a tagged customer in, first, the general queue and, then, the exponential queue. Subsequently, we indicate how the approach toward this closed system also allows us to study the joint sojourn time distribution of a tagged customer in the equivalent open two-queue system, consisting of FCFS single server queues with general and exponential service times, respectively, in the case that the input process to the first queue is a Poisson process.     

Limit non-stationary behavior of large closed queueing networks with bottlenecks

In this paper martingales methods are applied for analyzing limit non-stationary behavior of the queue length processes in closed Jackson queueing networks with a single class consisting of a large number of customers, a single infinite server queue, and a fixed number of single server queues with large state independent service rates. It is assumed that one of the single server nodes forms a bottleneck. For the non-bottleneck nodes we show that the queue length distribution at timet converges in generalized sense to the stationary distribution of the M/M/1 queue whose parameters explicitly depend ont. For the bottleneck node a diffusion approximation with reflection is proved in the moderate usage regime while fluid and Gaussian diffusion approximations are established for the heavy usage regime.     

On ergodicity conditions in a polling model with Markov modulated input and state-dependent routing

A polling system with switchover times and state-dependent server routing is studied. Input flows are modulated by a random external environment. Input flows are ordinary Poisson flows in each state of the environment, with intensities determined by the environment state. Service and switchover durations have exponential laws of probability distribution. A continuous-time Markov chain is introduced to describe the dynamics of the server, the sizes of the queues and the states of the environment. By means of the iterative-dominating method a sufficient condition for ergodicity of the system is obtained for the continuous-time Markov chain. This condition also ensures the existence of a stationary probability distribution of the embedded Markov chain at instants of jumps. The customers sojourn cost during the period of unloading the stable queueing system is chosen as a performance metric. Numerical study in case of two input flows and a class of priority and threshold routing algorithms is conducted. It is demonstrated that in case of light inputs a priority routing rule doesn’t seem to be quasi-optimal.     

Equilibrium customers’ choice between FCFS and random servers

Consider two servers of equal service capacity, one serving in a first-come first-served order (FCFS), and the other serving its queue in random order. Customers arrive as a Poisson process and each arriving customer observes the length of the two queues and then chooses to join the queue that minimizes its expected queueing time. Assuming exponentially distributed service times, we numerically compute a Nash equilibrium in this system, and investigate the question of which server attracts the greater share of customers. If customers who arrive to find both queues empty independently choose to join each queue with probability 0.5, then we show that the server with FCFS discipline obtains a slightly greater share of the market. However, if such customers always join the same queue (say of the server with FCFS discipline) then that server attracts the greater share of customers. This research was supported by the Israel Science Foundation grant No. 526/08.     

The MM CPP/GE/c G-Queue: Sojourn Time Distribution

We obtain the sojourn time probability distribution function at equilibrium for a Markov modulated, multi-server, single queue with generalised exponential (GE) service time distribution and compound Poisson arrivals of both positive and negative customers. Such arrival processes can model both burstiness and correlated traffic and are well suited to models of ATM and other telecommunication networks. Negative customers remove (ordinary) customers in the queue and are similarly correlated and bursty. We consider both the cases where negative customers remove positive customers from the front and the end of the queue and, in the latter case, where a customer currently being served can and cannot be killed by a negative customer. These cases can model an unreliable server or load balancing respectively. The results are obtained as Laplace transforms and can be inverted numerically. The MM CPP/GE/c G-Queue therefore holds the promise of being a viable building block for the analysis of queues and queueing networks with bursty, correlated traffic, incorporating load balancing and node-failures, since the equilibrium behaviour of both queue lengths and response times can be determined in a tractable way.     

Stability of a two-queue cyclic polling system with BMAPs under gated service and state-dependent time-limited service disciplines

The stability of a cyclic polling system, with a single server and two infinite-buffer queues, is considered. Customers arrive at the two queues according to independent batch Markovian arrival processes. The first queue is served according to the gated service discipline, and the second queue is served according to a state-dependent time-limited service discipline with the preemptive repeat-different property. The state dependence is that, during each cycle, the predetermined limited time of the server’s visit to the second queue depends on the queue length of the first queue at the instant when the server last departed from the first queue. The mean of the predetermined limited time for the second queue either decreases or remains the same as the queue length of the first queue increases. Due to the two service disciplines, the customers in the first queue have higher service priority than the ones in the second queue, and the service fairness of the customers with different service priority levels is also considered. In addition, the switchover times for the server traveling between the two queues are considered, and their means are both positive as well as finite. First, based on two embedded Markov chains at the cycle beginning instants, the sufficient and necessary condition for the stability of the cyclic polling system is obtained. Then, the calculation methods for the variables related to the stability condition are given. Finally, the influence of some parameters on the stability condition of the cyclic polling system is analyzed. The results are useful for engineers not only checking whether the given cyclic polling system is stable, but also adjusting some parameters to make the system satisfy some requirements under the condition that the system is stable.     

A network traffic flow model for motorway and urban highways

The research reported in this paper develops a network-level traffic flow model (NTFM) that is applicable for both motorways and urban roads. It forecasts the traffic flow rates, queue propagation at the junctions and travel delays through the network. NTFM uses sub-models associated with all road and junction types that comprise the highway. The flow at any one part of the network is obviously very dependent on the flows at all other parts of the network. To predict the two-way traffic flow in NTFM, an iterative simulation method is executed to generate the evolution of dependent traffic flows and queues. To demonstrate the capability of the model, it is applied to a small case study network and a local Loughborough–Nottingham highway network. The results indicate that NTFM is capable of identifying the relationship between traffic flows and capturing traffic phenomena such as queue dynamics. By introducing a reduced flow rate on links of the network, the effects of strategies used to carry out roadworks can be mimicked.     

Approximation of throughput in tandem queues with multiple servers and blocking

In this paper, we develop an approximation method for throughput in tandem queues with multiple independent reliable servers at each stage and finite buffers between service stations. We consider the blocking after service (BAS) blocking protocol of each service stage. The service time distribution of each server is exponential. The approximation is based on the decomposition of the system into a set of coupled subsystems which are modeled by two-stage tandem queue with two buffers and are analyzed by using the level dependent quasi-birth-and-death (LDQBD) process.