Sojourn time asymptotics in the M/G/1 processor sharing queue

We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index -ν, ν non-integer, iff the sojourn time distribution is regularly varying of index -ν. This result is derived from a new expression for the Laplace–Stieltjes transform of the sojourn time distribution. That expression also leads to other new properties for the sojourn time distribution. We show how the moments of the sojourn time can be calculated recursively and prove that the kth moment of the sojourn time is finite iff the kth moment of the service time is finite. In addition, we give a short proof of a heavy traffic theorem for the sojourn time distribution, prove a heavy traffic theorem for the moments of the sojourn time, and study the properties of the heavy traffic limiting sojourn time distribution when the service time distribution is regularly varying. Explicit formulas and multiterm expansions are provided for the case that the service time has a Pareto distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

The M/G/1 processor-sharing model: transient behavior

This paper deals with the M/G/1 model with processor-sharing service discipline. LetL ^* (t, x) denote the number of jobs present at timet whose attained service time is not greater thanx,x0, andV ₀(t,z) the sojourn time of a tagged job placed in the system at timet and requiringz units of service. Explicit analytical expressions are obtained for the joint distribution ofL ^*(t, ·) andV ⁰(t, ·) under various initial conditions in terms of the Laplace transform with respect tot. It is shown that for initial conditions of special kind (there is one job or none) the results can be expressed in a closed form.     

A sample path relation for the sojourn times in G/G/1−PS systems and its applications

For the single server system under processor sharing (PS) a sample path result for the sojourn times in a busy period is proved, which yields a sample path relation between the sojourn times under PS and FCFS discipline. This relation provides a corresponding one between the mean stationary sojourn times in G/G/1 under PS and FCFS. In particular, the mean stationary sojourn time in G/D/1 under PS is given in terms of the mean stationary sojourn time under FCFS, generalizing known results for GI/M/1 and M/GI/1. Extensions of these results suggest an approximation of the mean stationary sojourn time in G/GI/1 under PS in terms of the mean stationary sojourn time under FCFS. Mathematics Subject Classification (MSC 2000) 60K25· 68M20· 60G17· 60G10 This work was supported by a grant from the Siemens AG.     

Analysis of a three node queueing network

The three node Jackson queueing network is the simplest acyclic network in which in equilibrium the sojourn times of a customer at each of the nodes are dependent. We show that assuming the individual sojourn times are independent provides a good approximation to the total sojourn time. This is done by simulating the network and showing that the sojourn times generally pass a Kolmogorov-Smirnov test as having come from the approximating distribution. Since the sum of dependent random variables may have the same distribution as the sum of independent random variables with the same marginal distributions, it is conceivable that our approximation is exact. However, we numerically compute upper and lower bounds for the distribution of the total sojourn time; these bounds are so close that the approximating distribution lies outside of the bounds. Thus, the bounds are accurate enough to distinguish between the two distributions even though the Kolmogorov-Smirnov test generally cannot.     

Size- and state-aware dispatching problem with queue-specific job sizes

We consider the dispatching problem in a size- and state-aware multi-queue system with Poisson arrivals and queue-specific job sizes. By size- and state-awareness, we mean that the dispatcher knows the size of an arriving job and the remaining service times of the jobs in each queue. By queue-specific job sizes, we mean that the time to process a job may depend on the chosen server. We focus on minimizing the mean sojourn time (i.e., response time) by an MDP approach. First we derive the so-called size-aware relative values of states with respect to the sojourn time in an M/G/1 queue operating under FIFO, LIFO, SPT or SRPT disciplines. For FIFO and LIFO, the size-aware relative values turn out to be insensitive to the form of the job size distribution. The relative values are then exploited in developing efficient dispatching rules in the spirit of the first policy iteration.     

Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions

We consider a GI/G/1 queue in which the service time distribution and/or the interarrival time distribution has a heavy tail, i.e., a tail behaviour like t ^−ν with 1 < ν ⩽ 2 , so that the mean is finite but the variance is infinite. We prove a heavy-traffic limit theorem for the distribution of the stationary actual waiting time W. If the tail of the service time distribution is heavier than that of the interarrival time distribution, and the traffic load a → 1, then W, multiplied by an appropriate ‘coefficient of contraction’ that is a function of a, converges in distribution to the Kovalenko distribution. If the tail of the interarrival time distribution is heavier than that of the service time distribution, and the traffic load a → 1, then W, multiplied by another appropriate ‘coefficient of contraction’ that is a function of a, converges in distribution to the negative exponential distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

The M/M/c with critical jobs

We consider theM/M/c queue, where customers transfer to a critical state when their queueing (sojourn) time exceeds a random time. Lower and upper bounds for the distribution of the number of critical jobs are derived from two modifications of the original system. The two modified systems can be efficiently solved. Numerical calculations indicate the power of the approach.     

Heavy-tailed limits for medium size jobs and comparison scheduling

We study the conditional sojourn time distributions of processor sharing (PS), foreground background processor sharing (FBPS) and shortest remaining processing time first (SRPT) scheduling disciplines on an event where the job size of a customer arriving in stationarity is smaller than exactly k≥0 out of the preceding m≥k arrivals. Then, conditioning on the preceding event, the sojourn time distribution of this newly arriving customer behaves asymptotically the same as if the customer were served in isolation with a server of rate (1−ρ)/(k+1) for PS/FBPS, and (1−ρ) for SRPT, respectively, where ρ is the traffic intensity. Hence, the introduced notion of conditional limits allows us to distinguish the asymptotic performance of the studied schedulers by showing that SRPT exhibits considerably better asymptotic behavior for relatively smaller jobs than PS/FBPS. Inspired by the preceding results, we propose an approximation to the SRPT discipline based on a novel adaptive job grouping mechanism that uses relative size comparison of a newly arriving job to the preceding m arrivals. Specifically, if the newly arriving job is smaller than k and larger than m−k of the previous m jobs, it is routed into class k. Then, the classes of smaller jobs are served with higher priorities using the static priority scheduling. The good performance of this mechanism, even for a small number of classes m+1, is demonstrated using the asymptotic queueing analysis under the heavy-tailed job requirements. We also discuss refinements of the comparison grouping mechanism that improve the accuracy of job classification at the expense of a small additional complexity. This work is supported by NSF Grant 0615126.     

Sojourn Times in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">PH</Emphasis>/1 Processor Sharing Queue

We give in this paper an algorithm to compute the sojourn time distribution in the processor sharing, single server queue with Poisson arrivals and phase type distributed service times. In a first step, we establish the differential system governing the conditional sojourn times probability distributions in this queue, given the number of customers in the different phases of the PH distribution at the arrival instant of a customer. This differential system is then solved by using a uniformization procedure and an exponential of matrix. The proposed algorithm precisely consists of computing this exponential with a controlled accuracy. This algorithm is then used in practical cases to investigate the impact of the variability of service times on sojourn times and the validity of the so-called reduced service rate (RSR) approximation, when service times in the different phases are highly dissymmetrical. For two-stage PH distributions, we give conjectures on the limiting behavior in terms of an M/M/1 PS queue and provide numerical illustrative examples.This revised version was published online in June 2005 with corrected coverdate     

Additive functionals with application to sojourn times in infinite-server and processor sharing systems

We deal with additive functionals of stationary processes. It is shown that under some assumptions a stationary model of the time-changed process exists. Further, bounds for the expectation of functions of additive functionals are derived. As an application we analyze virtual sojourn times in an infinite-server system where the service speed is governed by a stationary process. It turns out that the sojourn time of some kind of virtual requests equals in distribution an additive functional of a stationary time-changed process, which provides bounds for the expectation of functions of virtual sojourn times, in particular bounds for fractional moments and the distribution function. Interpreting the GI(n)/GI(n)/∞ system or equivalently the GI(n)/GI system under state-dependent processor sharing as an infinite-server system where the service speed is governed by the number n of requests in the system provides results for sojourn times of virtual requests. In the case of M(n)/GI(n)/∞, the sojourn times of arriving and added requests equal in distribution sojourn times of virtual requests in modified systems, which yields several results for the sojourn times of arriving and added requests. In case of positive integer moments, the bounds generalize earlier results for M/GI(n)/∞. In particular, the mean sojourn times of arriving and added requests in M(n)/GI(n)/∞ are proportional to the required service time, generalizing Cohen’s famous result for M/GI(n)/∞.     

Large deviations of sojourn times in processor sharing queues

This paper presents a large deviation analysis of the steady-state sojourn time distribution in the GI/G/1 PS queue. Logarithmic estimates are obtained under the assumption of the service time distribution having a light tail, thus supplementing recent results for the heavy-tailed setting. Our proof gives insight into the way a large sojourn time occurs, enabling the construction of an (asymptotically efficient) importance sampling algorithm. Finally our results for PS are compared to a number of other service disciplines, such as FCFS, LCFS, and SRPT. 2000 mathematics subject classification: 60K25.     

Optimal choice of threshold in Two Level Processor Sharing

We analyze the Two Level Processor Sharing (TLPS) scheduling discipline with the hyper-exponential job size distribution and with the Poisson arrival process. TLPS is a convenient model to study the benefit of the file size based differentiation in TCP/IP networks. In the case of the hyper-exponential job size distribution with two phases, we find a closed form analytic expression for the expected sojourn time and an approximation for the optimal value of the threshold that minimizes the expected sojourn time. In the case of the hyper-exponential job size distribution with more than two phases, we derive a tight upper bound for the expected sojourn time conditioned on the job size. We show that when the variance of the job size distribution increases, the gain in system performance increases and the sensitivity to the choice of the threshold near its optimal value decreases. The work was supported by France Telecom R&D Grant “Modélisation et Gestion du Trafic Réseaux Internet” no. 46129414.     

Analysis of preemptive loss priority queues with preemption distance

In a queueing system with preemptive loss priority discipline, customers disappear from the system immediately when their service is preempted by the arrival of another customer with higher priority. Such a system can model a case in which old requests of low priority are not worthy of deferred service. This paper is concerned with preemptive loss priority queues in which customers of each priority class arrive in a Poisson process and have general service time distribution. The strict preemption in the existing model is extended by allowing the preemption distance parameterd such that arriving customers of only class 1 throughp — d can preempt the service of a customer of classp. We obtain closed-form expressions for the mean waiting time, sojourn time, and queue size from their distributions for each class, together with numerical examples. We also consider similar systems with server vacations.     

On the average sojourn time under M/M/1/SRPT

We study an M/M/1 queueing system under the shortest remaining processing time (SRPT) policy. We show that the average sojourn time varies as , where ρ is the system load. Thus, SRPT offers a Θ(ln(e/(1−ρ))) factor improvement over policies that ignore knowledge of job sizes while scheduling.     

Insensitivity of multiclass systems with general dynamic preemptive resume queueing disciplines

We are concerned with the insensitivity of the stationary distributions of the system states inM/G/s/m queues with multiclass customers and with LIFO preemptive resume service disciplines. We introduce general entrance and exit rules into and from waiting positions, respectively, for the behaviour of waiting customers whose service is interrupted. These rules may, roughly speaking, depend on the number of customers in the system. It is shown that the stationary distribution of the system state is insensitive not only with respect to the service time distributions but also with respect to the general entrance and exit rules. As well as the insensitivity of the service scheme, our results are obtained for a special form of state and customer type dependent arrival and service rates. Some further results are concluded related to insensitivity like the formula for the conditional mean sojourn time and the property of transformation of a Poisson input into a Poisson output by the systems.     

TheM/G/1 queue with processor sharing and its relation to a feedback queue

The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1–p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.Some variants are also considered, viz., anM/M/1 feedback queue with additional customers who are always present, and anM/G/1 processor sharing queue with feedback.     

Some results on a generalized M/G/1 feedback queue with negative customers

This paper deals with a generalized M/G/1 feedback queue in which customers are either “positive" or “negative". We assume that the service time distribution of a positive customer who initiates a busy period is G _e (x) and all subsequent positive customers in the same busy period have service time drawn independently from the distribution G _b (x). The server is idle until a random number N of positive customers accumulate in the queue. Following the arrival of the N-th positive customer, the server serves exhaustively the positive customers in the queue and then a new idle period commences. This queueing system is a generalization of the conventional N-policy queue with N a constant number. Explicit expressions for the probability generating function and mean of the system size of positive customers are obtained under steady-state condition. Various vacation models are discussed as special cases. The effects of various parameters on the mean system size and the probability that the system is empty are also analysed numerically. AMS Subject Classification: Primary: 60 K 25 · Secondary: 60 K 20, 90 B 22     

Queue length and waiting time of the M/G/1 queue under the D-policy and multiple vacations

We study the steady-state queue length and waiting time of the M/G/1 queue under the D-policy and multiple server vacations. We derive the queue length PGF and the LSTs of the workload and waiting time. Then, the mean performance measures are derived. Finally, a numerical example is presented and the effects of employing the D-policy are discussed. AMS Subject Classifications 60K25 This work was supported by the SRC/ERC program of MOST/KOSEF grant # R11-2000-073-00000.     

Heavy-traffic analysis of the sojourn time in a three node Jackson network with overtaking

We consider a Jackson network consisting of three first-in-first-out (FIFO)M/M/1 queues. When customers leave the first queue they can be routed to either the second or third queue. Thus, a customer that traverses the network by going from the first to the second to the third queue, can be overtaken by another customer that is routed from the first queue directly to the third. We study the distribution of the sojourn time of a customer through the three node network, in the heavy traffic limit. A three term heavy traffic asymptotic approximation to the sojourn time density is derived. The leading term shows that the nodes decouple in the heavy traffic limit. The next two terms, however, do show the dependence of the sojourn times at the individual nodes and give quantitative measures of the effects of overtaking.     

A decomposition theorem and related results for the discriminatory processor sharing queue

In this paper, we study a discriminatory processor sharing queue with Poisson arrivals,K classes and general service times. For this queue, we prove a decomposition theorem for the conditional sojourn time of a tagged customer given the service times and class affiliations of the customers present in the system when the tagged customer arrives. We show that this conditional sojourn time can be decomposed inton+1 components if there aren customers present when the tagged customer arrives. Further, we show that thesen+1 components can be obtained as a solution of a system of non-linear integral equations. These results generalize known results about theM/G/1 egalitarian processor sharing queue.