Age of Job Completion Minimization with Stable Queues

TL;DR

This work addresses minimizing the age of job completion plus sampling cost in a time-slotted, multi-user offloading system with a Markovian machine. It introduces the age of job completion as a key metric and develops two policy families: adaptive randomized scheduling/sampling and a max-age scheduling with adaptive sampling, providing closed-form performance expressions for stationary policies on fixed user subsets and deriving sufficient queue-stability conditions. Through numerical evaluation, the max-age policy consistently outperforms the adaptive randomized approach, particularly at higher arrival rates, demonstrating the practical viability of these strategies for edge-computing offloading with uncertain device dynamics. The results offer design guidelines for balancing timely job completion against state-sampling costs while ensuring queue stability in stochastic, Markov-driven environments.

Abstract

We consider a time-slotted job-assignment system with a central server, N users and a machine which changes its state according to a Markov chain (hence called a Markov machine). The users submit their jobs to the central server according to a stochastic job arrival process. For each user, the server has a dedicated job queue. Upon receiving a job from a user, the server stores that job in the corresponding queue. When the machine is not working on a job assigned by the server, the machine can be either in internally busy or in free state, and the dynamics of these states follow a binary symmetric Markov chain. Upon sampling the state information of the machine, if the server identifies that the machine is in the free state, it schedules a user and submits a job to the machine from the job queue of the scheduled user. To maximize the number of jobs completed per unit time, we introduce a new metric, referred to as the age of job completion. To minimize the age of job completion and the sampling cost, we propose two policies and numerically evaluate their performance. For both of these policies, we find sufficient conditions under which the job queues will remain stable.

Figures (4)

• Figure 1: A job-assignment system with a Markov machine, a central server and $N$ users.
• Figure 2: In this figure, we consider a system with $N=2$ users, and we show the corresponding state transitions of the Markov machine. The transitions labeled with black arrows represent slot-by-slot transitions, whereas those labeled with green, grey, orange, and pink arrows denote instantaneous transitions resulting from the server actions, namely scheduling and job assignment. When the machine is in the free state while both job queues are empty, and the server samples the state, the machine remains free, and we denote this self-transition with the green arrow. Similarly, when the machine is in the free state, and the server samples and assigns a job from the job-queue of the first (second) user to the machine, the machine instantaneously transitions to state $1$ ($2$), denoted by the pink (orange) arrow.
• Figure 3: We consider a system with $N=2$ users. The figure illustrates a sample path of age of job completion and the corresponding job queue length for the first user under an arbitrary policy $\phi$. At each time, we color the age block based on the current state of the machine. We note that a job arrives at the end of a time slot which we denote with blue arrows, and the server performs sampling and job assignment at the beginning of a slot with colored arrows following the same color conventions as in Fig. \ref{['fig:1']}. For example, during time slot $3$, the machine is internally busy and transitions to the free state at the end of the slot. At the beginning of slot $4$, the server samples the state of the machine and schedules a job from the job queue of user $2$ instantaneously.
• Figure 4: Total average cost of policies $\phi_1$ and $\bar{\phi}_1$ as a function of $q$, under two distinct arrival rate configurations, $\bm{p}$ and $\tilde{\bm{p}}$, for a system of $N=4$ users.

Theorems & Definitions (13)

• Definition 1
• Remark 1
• Definition 2
• Definition 3
• Remark 2
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 1