Exponential Server Queues Research Articles

Degradedness and Secrecy in Memoryless Queues

We study exponential-server timing channels, in which the transmitter encodes a message using a sequence of packet inter-arrival times and the receiver observes the corresponding inter-departure times from an exponential-server queue. We show that if the service rate of one such queue is faster than that of another, then the latter channel is stochastically degraded with respect to the former. This combined with Burke’s theorem implies that the slower queue is more entropy increasing than the first. These facts put together provide a converse for the secrecy capacity of the wiretapped version of the channel, matching an earlier achievability result in the literature.

Asymptotic independence of queues under randomized load balancing

Randomized load balancing greatly improves the sharing of resources while being simple to implement. In one such model, jobs arrive according to a rate-?N Poisson process, with ?<1, in a system of N rate-1 exponential server queues. In Vvedenskaya et al. (Probl. Inf. Transm. 32:15---29, 1996), it was shown that when each arriving job is assigned to the shortest of D, D?2, randomly chosen queues, the equilibrium queue sizes decay doubly exponentially in the limit as N??. This is a substantial improvement over the case D=1, where queue sizes decay exponentially. The reasoning in Vvedenskaya et al. (Probl. Inf. Transm. 32:15---29, 1996) does not easily generalize to jobs with nonexponential service time distributions. A modularized program for treating randomized load balancing problems with general service time distributions was introduced in Bramson et al. (Proc. ACM SIGMETRICS, pp. 275---286, 2010). The program relies on an ansatz that asserts that, for a randomized load balancing scheme in equilibrium, any fixed number of queues become independent of one another as N??. This allows computation of queue size distributions and other performance measures of interest. In this article, we demonstrate the ansatz in several settings. We consider the least loaded balancing problem, where an arriving job is assigned to the queue with the smallest workload. We also consider the more difficult problem, where an arriving job is assigned to the queue with the fewest jobs, and demonstrate the ansatz when the service discipline is FIFO and the service time distribution has a decreasing hazard rate. Last, we show the ansatz always holds for a sufficiently small arrival rate, as long as the service distribution has 2 moments.

Randomized load balancing with general service time distributions

Randomized load balancing greatly improves the sharing of resources in a number of applications while being simple to implement. One model that has been extensively used to study randomized load balancing schemes is the supermarket model. In this model, jobs arrive according to a rate-nλ Poisson process at a bank of n rate-1 exponential server queues. A notable result, due to Vvedenskaya et.al. (1996), showed that when each arriving job is assigned to the shortest of d ≥ 2 randomly chosen queues, the equilibrium queue sizes decay doubly exponentially in the limit as n to ∞. This is a substantial improvement over the case d=1, where queue sizes decay exponentially. The method of analysis used in the above paper and in the subsequent literature applies to jobs with exponential service time distributions and does not easily generalize. It is desirable to study load balancing models with more general, especially heavy-tailed, service time distributions since such service times occur widely in practice. This paper describes a modularized program for treating randomized load balancing problems with general service time distributions and service disciplines. The program relies on an ansatz which asserts that any finite set of queues in a randomized load balancing scheme becomes independent as n to ∞. This allows one to derive queue size distributions and other performance measures of interest. We establish the ansatz when the service discipline is FIFO and the service time distribution has a decreasing hazard rate (this includes heavy-tailed service times). Assuming the ansatz , we also obtain the following results: (i) as n to ∞, the process of job arrivals at any fixed queue tends to a Poisson process whose rate depends on the size of the queue, (ii) when the service discipline at each server is processor sharing or LIFO with preemptive resume, the distribution of the number of jobs is insensitive to the service distribution, and (iii) the tail behavior of the queue-size distribution in terms of the service distribution for the FIFO service discipline.

The Logistics Impact of a Mixture of Order-Streams in a Manufacturer-Retailer System

We model a supply chain with two retail warehouses that place replenishment orders with a common manufacturing capacity. The two retailers differ in the variability of their order-streams. The order-stream from one retail warehouse is modeled as a Poisson process and from the other as a hyperexponential renewal process. Each retail warehouse uses a base-stock policy to place replenishment orders with the manufacturer. The manufacturer is modeled as a first-come-first-serve, single exponential server queue. We analyze the supply-side impact of this mixture of order-streams received by the manufacturer on both retailers. An exact analysis of this base-model generates closed-form expressions for distributions of the lead-time, outstanding orders, and expected inventory costs for each retailer, and leads to comparative results about the two retailers- performance measures. The base-model is extended to accommodate finished goods at the manufacturer, more than two retailers, and bulk-arrivals. We use the model to suggest managerial insights about the impact of the presence of a high-variability retailer on other retailers who share capacity, the distorting impact of manufacturer finished goods inventory on retailer incentives, and the incentives for retailers to participate in variability-reduction programs in the grocery industry.

Entropy and the timing capacity of discrete queues

Queueing systems which map Poisson input processes to Poisson output processes have been well-studied in classical queueing theory. This paper considers two discrete-time queues whose analogs in continuous-time possess the Poisson-in-Poisson-out property. It is shown that when packets arriving according to an arbitrary ergodic stationary arrival process are passed through these queueing systems, the corresponding departure process has an entropy rate no less (some times strictly more) than the entropy rate of the arrival process. Some useful by-products are discrete-time versions of: (i) a proof of the celebrated Burke's (1956) theorem, (ii) a proof of the uniqueness, amongst renewal inputs, of the Poisson process as a fixed point for exponential server queues proposed by Anantharam (1993), and (iii) connections with the timing capacity of queues described by Anantharam and Verdu (1996).

Sequential decoding for the exponential server timing channel

We show the existence of a good tree code with a sequential decoder for the exponential server timing channel. The expected number of computations before moving one step ahead is upper-bounded by a finite number. The rate of information transfer for this code is /spl mu//(2e) nats per second i.e., one half of the capacity. The cutoff rate for the exponential server queue is therefore at least /spl mu//(2u) nats per second.

Convergence of Departures in Tandem Networks of·/GI/ ∞ Queues

We consider an infinite series of independent and identical -/GI/∞ queues fed by an arbitrary stationary and ergodic arrival process, A1. Let Ai be the arrival process to the ith node, and let νi be the law of Ai. Denote by (·) the input–output map of the ·/GI/∞ node; that is, νi+1 = (νi). It is known that the Poisson process is a fixed point for . In this paper, we are interested in the asymptotic distribution of the departure process from the nth node, νn+1 = n(ν1), as n ∞. Using couplings for random walks, this limiting distribution is shown to be either a Poisson process or a stationary ν-Poisson process, depending on the joint distribution of A1 and the service process. This generalizes a result of Vere-Jones (1968, Journal of the Royal Statistical Society, Series B 30: 321–333) and is similar in flavor to Mountford and Prabhakar (1995, Annals of Applied Probability 5(1): 121–127), where Poisson convergence is established for departures from a series of exponential server queues using coupling methods.

Stationary tail probabilities in exponential server tandems with renewal arrivals

The problem considered is that of estimating the tail stationary probability for two exponential server queues in series fed by renewal arrivals. We compute the tail of the marginal queue length distribution at the second queue. The marginal at the first queue is known by the classical result for the GI/M/1 queue. The approach involves deriving necessary and sufficient conditions on the paths of the arrival and virtual service processes in order to get a large queue size at the second queue. We then use large deviations estimates of the probabilities of these paths, and solve a constrained convex optimization problem to find the most likely path leading to a large queue size. We find that the stationary queue length distribution at the second queue has an exponentially decaying tail, and obtain the exact rate of decay.

On the Weak Convergence of Departures from an Infinite Series of $\cdot /M/ 1$ Queues

In this note we observe that the recent argument of Ekhaus and Gray combined with the approach of Liggett and Shiga shows that the limit from passing a stationary ergodic arrival process of rate $\alpha < 1$ through a sequence of independent, rate one, exponential server queues is a Poisson process of rate $\alpha$. This builds on work of Liggett and Shiga and Anantharam.

Further properties of extremal sequences in queues

In a recent paper by Hajek (1985), an extremal zero-one valued sequence of asymptotic mean p generated by [(n+1)]-[np] is considered. It has been shown there that if a fraction p of customers from a renewal process is sent to an exponential server queue according to a pre-specified splitting sequence, then the long-term average queue size seen by routed and unrouted customers is minimized when the extremal sequence is used. In this paper, further properties of extremal sequences which enable one to study the optimal splitting of customers in more general queues are investigated. It is shown, for example, that the extremal sequence is superior to any other sequences in the sense of convex ordering for the waiting time of an arbitrary routed customer in GI/GI/l queues under FCFS discipline, thereby enhancing the importance of extremal splittings in practice.

Extremal Splittings of Point Processes

The sequence with nth term defined by [(n + 1)p] − [np] is an extremal zero-one valued sequence of asymptotic mean p in the following sense (for example): if a fraction p of customers from a point process with iid interarrival times is sent to an exponential server queue according to a prespecified splitting sequence, then the long-term average queue size is minimized when the above sequence is used. The proof involves consideration of the lower convex envelope J (which is a function on Rm) of a function J on Zm. An explicit representation is given for J in terms of J, for J in a broad class of functions, which we call “multimodular.” The expected queue size just before an arrival, considered as a function of the zero-one splitting sequence, is shown to belong to this class.

Finite Capacity Queuing Systems with Applications in Computer Modeling

A queuing system with a buffer of unlimited capacity in front of a cyclic arrangement of two exponential server queues is analyzed. The main feature of the system is blocking, i.e., when the population in the two queues attains a maximum value M, say, new arrivals are held back in the buffer. The solution is given in form of polynomial equations which require the roots of a characteristic equation. A solution algorithm is provided. The stability condition is given in terms of these roots and also in explicit form. Limiting cases which are of practical interest are discussed. These limiting cases lead to a better understanding of some popular approximation techniques.