Yiqiang Q. Zhao

Qihui Bu, Yiqiang Q. Zhao

In 1985, Grassmann, Taksar, and Heyman published their celebrated paper, in which they introduced a numerically stable algorithm for computing the stationary probabilities of a finite-state Markov chain, one of the key performance quantities in both theory and applications. This algorithm later became the well-known GTH algorithm (or the state-reduction method) in the literature, becoming one of the standard algorithms in applied probability. Later, this algorithm was extended to deal with the stationary distributions of block-structured Markov chains with repeating rows. In this paper, we focus on the block-form GTH algorithm and organize it into two parts. In the first part, we connect the block-form GTH algorithm to censored Markov chains and the block-form $RG$-factorization. We show that the forward block-elimination and back block-form substitution of the block-form GTH algorithm are equivalent to solving a system formulated using the $RG$-factorization in two steps. We also show that this connection remains valid when the block-form GTH algorithm is extended to infinite-state Markov chains. It is well known that censoring an infinite-state Markov chain to a finite state space yields a stationary distribution that provides a best approximation to the stationary distribution of the original infinite-state Markov chain. In the second part, we first derive an explicit expression for the censored Markov chain from the infinite state space to a finite space for Markov chains of $M/G/1$ type. Based on this expression, we propose a renormalized approximated censored transition matrix (RA-CM). The resulting stationary distribution is shown to be asymptotically optimal in terms of approximation error. We compare the approximation error of the RA-CM with the error arising from the last-block-column augmentation.

Ahmed Sid-Ali, Ioannis Lambadaris, Yiqiang Q. Zhao et al.

We tackle in this paper an online network resource allocation problem with job transfers. The network is composed of many servers connected by communication links. The system operates in discrete time; at each time slot, the administrator reserves resources at servers for future job requests, and a cost is incurred for the reservations made. Then, after receptions, the jobs may be transferred between the servers to best accommodate the demands. This incurs an additional transport cost. Finally, if a job request cannot be satisfied, there is a violation that engenders a cost to pay for the blocked job. We propose a randomized online algorithm based on the exponentially weighted method. We prove that our algorithm enjoys a sub-linear in time regret, which indicates that the algorithm is adapting and learning from its experiences and is becoming more efficient in its decision-making as it accumulates more data. Moreover, we test the performance of our algorithm on artificial data and compare it against a reinforcement learning method where we show that our proposed method outperforms the latter.

Ahmed Sid-Ali, Ioannis Lambadaris, Yiqiang Q. Zhao et al.

This paper studies an online optimal resource reservation problem in communication networks with job transfers where the goal is to minimize the reservation cost while maintaining the blocking cost under a certain budget limit. To tackle this problem, we propose a novel algorithm based on a randomized exponentially weighted method that encompasses long-term constraints. We then analyze the performance of our algorithm by establishing an upper bound for the associated regret and the cumulative constraint violations. Finally, we present numerical experiments where we compare the performance of our algorithm with those of reinforcement learning where we show that our algorithm surpasses it.

Ahmed Sid-Ali, Ioannis Lambadaris, Yiqiang Q. Zhao et al.

In this paper, we study an optimal online resource reservation problem in a simple communication network. The network is composed of two compute nodes linked by a local communication link. The system operates in discrete time; at each time slot, the administrator reserves resources for servers before the actual job requests are known. A cost is incurred for the reservations made. Then, after the client requests are observed, jobs may be transferred from one server to the other to best accommodate the demands by incurring an additional transport cost. If certain job requests cannot be satisfied, there is a violation that engenders a cost to pay for each of the blocked jobs. The goal is to minimize the overall reservation cost over finite horizons while maintaining the cumulative violation and transport costs under a certain budget limit. To study this problem, we first formalize it as a repeated game against nature where the reservations are drawn randomly according to a sequence of probability distributions that are derived from an online optimization problem over the space of allowable reservations. We then propose an online saddle-point algorithm for which we present an upper bound for the associated K-benchmark regret together with an upper bound for the cumulative constraint violations. Finally, we present numerical experiments where we compare the performance of our algorithm with those of simple deterministic resource allocation policies.