Aleksi Penttinen

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.

M/M/1-PS queue and size-aware task assignment

We consider a distributed server system in which heterogeneous servers operate under the processor sharing (PS) discipline. Exponentially distributed jobs arrive to a dispatcher, which assigns each task to one of the servers. In the so-called size-aware system, the dispatcher is assumed to know the remaining service requirements of some or all of the existing jobs in each server. The aim is to minimize the mean sojourn time, i.e., the mean response time. To this end, we first analyze an M/M/1-PS queue in the framework of Markov decision processes, and derive the so-called size-aware relative value of state, which sums up the deviation from the average rate at which sojourn times are accumulated in the infinite time horizon. This task turns out to be non-trivial. The exact analysis yields an infinite system of first order differential equations, for which an explicit solution is derived. The relative values are then utilized to develop efficient dispatching policies by means of the first policy iteration (FPI). Numerically, we show that for the exponentially distributed job sizes the myopic approach, ignoring the future arrivals, yields an efficient and robust policy when compared to other heuristics. However, in the case of highly asymmetric service rates, an FPI based policy outperforms it. Additionally, the size-aware relative value of an M/G/1-PS queue is shown to be sensitive with respect to the form of job size distribution, and indeed, the numerical experiments with constant job sizes confirm that the optimal decision depends on the job size distribution.

Non-myopic vehicle and route selection in dynamic DARP with travel time and workload objectives

In dial-a-ride problems, a fleet of n vehicles is routed to transport people between pick-up and delivery locations. We consider an elementary version of the problem where trip requests arrive in time and require an immediate vehicle assignment (which triggers an appropriate route update of the selected vehicle). In this context, a relatively general objective can be stated as a weighted sum of the systems effort and the customers inconvenience. However, optimizing almost any objective in this immensely complex stochastic system is prohibitively difficult. Thus the earlier work has largely resorted to heuristic cost functions that arise, e.g., from the corresponding static systems. By using the framework of Markov decision processes and the classical M/M/1 queue as a highly abstract model for a single vehicle, we explain why certain intuitive cost functions indeed give satisfactory results in the dynamic system, and also give an explicit interpretation of different components appearing in a general cost function. The resulting family of heuristic control policies is demonstrated to offer a desired type of performance thus justifying the assumed analogy between a multi-queue and dial-a-ride systems.

On the optimal trade-off between SRPT and opportunistic scheduling

We consider service systems where new jobs not only increase the load but also improve the service ability of such a system, cf. opportunistic scheduling gain in wireless systems. We study the optimal trade-off between the SRPT (Shortest Remaining Processing Time) discipline and opportunistic scheduling in the systems characterized by compact and symmetric capacity regions. The objective is to minimize the mean delay in a transient setting where all jobs are available at time 0 and no new jobs arrive thereafter. Our main result gives conditions under which the optimal rate vector does not depend on the sizes of the jobs as long as their order (in size) remains the same. In addition, it shows that in this case the optimal policy applies the SRPT principle serving the shortest job with the highest rate of the optimal rate vector, the second shortest with the second highest rate etc. We also give a recursive algorithm to determine both the optimal rate vector and the minimum mean delay. In some special cases, the rate vector, as well as the minimum mean delay, have even explicit expressions as demonstrated in the paper. For the general case, we derive both an upper bound and a lower bound of the minimum mean delay.

Energy-aware dispatching in parallel queues with on-off energy consumption

We consider a dynamic dispatching problem where jobs are assigned upon arrival into parallel queues. Jobs have an arbitrary size distribution and each queue has its own service rate, queueing discipline, and operating power while serving jobs. Our goal is to minimize a weighted sum of delay and energy consumption under the assumption that the dispatcher is aware of the remaining service time of each job in the system, including that of the arriving job. We devise efficient dispatching heuristics based on the first policy iteration procedure of Markov Decision Processes. The resulting policies are illustrated by numerical examples. Direct control over the trade-off between performance and energy consumption will be increasingly important in future ICT equipment designs wherever dynamic queue assignment is needed.

Minimizing slowdown in heterogeneous size-aware dispatching systems

We consider a system of parallel queues where tasks are assigned (dispatched) to one of the available servers upon arrival. The dispatching decision is based on the full state information, i.e., on the sizes of the new and existing jobs. We are interested in minimizing the so-called mean slowdown criterion corresponding to the mean of the sojourn time divided by the processing time. Assuming no new jobs arrive, the shortest-processing-time-product (SPTP) schedule is known to minimize the slowdown of the existing jobs. The main contribution of this paper is three-fold: 1) To show the optimality of SPTP with respect to slowdown in a single server queue under Poisson arrivals; 2) to derive the so-called size-aware value functions for M/G/1-FIFO/LIFO/SPTP with general holding costs of which the slowdown criterion is a special case; and 3) to utilize the value functions to derive efficient dispatching policies so as to minimize the mean slowdown in a heterogeneous server system. The derived policies offer a significantly better performance than e.g., the size-aware-task-assignment with equal load (SITA-E) and least-work-left (LWL) policies.

Simulation of a large scale dynamic pickup and delivery problem

We study a variant of dynamic vehicle routing problem with pickups and deliveries where a vehicle is allocated to each service (i.e., trip) request immediately upon the arrival of the request. Solutions to this problem can be characterized as dynamic policies that define how each customer is handled by operating a fleet of vehicles. Evaluation of such policies is beyond the grasp of analytical studies and requires extensive simulations. We present an efficient and modular simulation tool developed for studying the performance of a large scale system with different policies under given trip arrival process. Numerical and analytical observations on the model are utilized to provide guidelines for solving the routing problem efficiently, and to support the validation of the simulation results. Application of the developed framework is demonstrated by several numerical examples, e.g., policy parameter optimization, which all give insight on the viability of this type of transportation system.

Optimal size-based opportunistic scheduler for wireless systems

Modern wireless cellular systems are able to utilize the opportunistic scheduling gain originating from the variability in the users’ channel conditions. By favoring users with good instantaneous channel conditions, the service capacity of the system can be increased with the number of users. On the other hand, for service systems with fixed service capacity, the system performance can be optimized by utilizing the size information. Combining the advantages of size-based scheduling with opportunistic scheduling gain has proven to be a challenging task. In this paper, we consider scheduling of data traffic (finite-size elastic flows) in wireless cellular systems. Assuming that the channel conditions for different users are independent and identically distributed, we show how to optimally combine opportunistic and size-based scheduling in the transient setting with all flows available at time 0. More specifically, by utilizing the time scale separation assumption, we develop a recursive algorithm that produces the optimal long-run service rate vectors within the corresponding capacity regions. We also prove that the optimal operating policy applies the SRPT-FM principle, i.e., the shortest flow is served with the highest rate of the optimal rate vector, the second shortest with the second highest rate, etc. Moreover, we determine explicitly how to implement the optimal rate vectors in the actual time slot level opportunistic scheduler. In addition to the transient setting, we explore the dynamic case with randomly arriving flows under illustrative channel scenarios by simulations. Interestingly, the scheduling policy that is optimal for the transient setting can be improved in the dynamic case under high traffic load by applying a rate-based priority scheduler that breaks the ties based on the SRPT principle.

Flow-level modeling and analysis of dynamic TDD in LTE

We present a queueing analysis of elastic traffic performance in LTE systems using the dynamic TDD scheme. Both fair resource sharing and performance optimization using different approaches are considered. We first analyze the system without any restrictions in the resource allocation between the uplink and downlink and demonstrate that a simple dynamic scheduling scheme (called here dynamic-PS), where the allocation for a single flow is always inversely proportional to the total number of active flows, shows good performance and fairness properties compared with any optimized static allocation scheme. We also consider the achievable gains with more detailed traffic statistics, including the application of the Gittins index policy and SRPT. However, the actual LTE TDD system only supports a discrete set of possible allocations in the capacity region. We then investigate how these allocation constraints impact the performance of the discretized variant of the dynamic-PS policy by using different approaches. To optimize the performance we apply MDP for exponential service times and, for example, derive a structural result that the optimal policy always selects among two corner points of the capacity region. Also, an SRPT-like heuristic scheduling algorithm is given. The analytical and simulated results suggest that the discrete dynamic-PS policy is robust against impact of different service time distributions, fair and performs reasonably well.

Congestive collapse and its avoidance in a dynamic dial-a-ride system with time windows

In a dynamic dial-a-ride problem (DARP) the task is to provide a transportation service in a given area by dynamically routing a set of vehicles in response to passengers trip requests. Passengers share vehicles similarly as with buses, while the schedule and routes are chosen ad hoc. Each trip is defined by the origin-destination pair in plane augmented with a latest feasible delivery time. Optimal control of such a system is a complicated task in general and outside the scope of this paper. Instead, we consider a set of well-defined heuristic control policies that can be evaluated by means of simulations. The main contribution of this paper is two-fold: (i) to demonstrate that a phenomenon known as congestive collapse occurs as the rate of trip requests increases beyond a capacity threshold of the given control policy (the value of which itself is unknown a priori); (ii) to propose a robust and computationally lightweight countermeasure to avoid the congestive collapse in such a way that the systems performance still improves after the capacity threshold has been passed. Despite its appealing simplicity, the proposed method succeeds in rejecting customers detrimental for the common good.