Asymptotically optimal open-loop load balancing

Abstract

In many distributed computing systems, stochastically arriving jobs need to be assigned to servers with the objective of minimizing waiting times. Many existing dispatching algorithms are basically included in the SQ(d) framework: Upon arrival of a job, $d\ge 2$ servers are contacted uniformly at random to retrieve their state and then the job is routed to a server in the best observed state. One practical issue in this type of algorithm is that server states may not be observable, depending on the underlying architecture. In this paper, we investigate the assignment problem in the open-loop setting where no feedback information can flow dynamically from the queues back to the controller, i.e., the queues are unobservable. This is an intractable problem, and unless particular cases are considered, the structure of an optimal policy is not known. Under mild assumptions and in a heavy-traffic many-server limiting regime, our main result proves the optimality of a subset of deterministic and periodic policies within a wide set of (open-loop) policies that can be randomized or deterministic and can be dependent on the arrival process at the controller. The limiting value of the scaled stationary mean waiting time achieved by any policy in our subset provides a simple approximation for the optimal system performance.

Appendix: Proof of Proposition 3

Since $p_r = \mu _r \bar{v}$ is a constant and $\frac{ \lambda _k k p_r}{\mu _r\Vert p\Vert (k+f_k)}<1$, we can directly use (54) to obtain

$$\begin{aligned} \overline{W}^{(k)}(\pi ^{(k)}) \le&R\frac{k+f_k}{\lambda _k k} + \sum _{r=1}^R \frac{\mu _r}{\Vert \mu \Vert } \, \mathcal {U}_r^{(k)}\left( \frac{\mu _r}{\Vert \mu \Vert (k+f_k)} \right) , \end{aligned}$$

(59a)

for any $\pi ^{(k)}\in \mathcal {A}_{\bar{v}\mu -TTRR}^{(k)}$. Thus,

$$\begin{aligned} \limsup _{k\rightarrow \infty }\frac{f_k}{k} \overline{W}^{(k)}(\pi ^{(k)}) \le&\sum _{r=1}^R \frac{\mu _r^2}{2\Vert \mu \Vert } \, \limsup _{k\rightarrow \infty } \frac{f_k}{k} \frac{ \sigma _r^2 + \varsigma \frac{\Vert \mu \Vert }{\mu _r} }{ 1- \frac{\lambda _k k}{\Vert \mu \Vert (k+f_k)} } = \sum _{r=1}^R \frac{\mu _r^2 \bar{\sigma }_r^2 }{2\Vert \mu \Vert }. \end{aligned}$$

(60a)

On the other hand, using Lemma 1 when q is given by $q_{r,k}=\mu _r/(\Vert \mu \Vert (k+f_k))$, we obtain

$$\begin{aligned} \inf _{\pi \in \mathcal {A}_{q}^{(k)}} \underline{W}^{(k)}(\pi )\ge & {} \sum _{r=1}^R\sum _{\kappa =1}^{k+f_k} q_{r,\kappa } \mathbb {E}\mathcal {W}_{r,\kappa }^{(k)}(\underline{V}(q_{r,\kappa }))\\= & {} \sum _{r=1}^R \frac{\mu _r}{\Vert \mu \Vert } \mathbb {E}\mathcal {W}_{r,\kappa }^{(k)}\left( \underline{V}\left( \frac{\mu _r}{\Vert \mu \Vert (k+f_k)}\right) \right) . \end{aligned}$$

Thus, as desired, we obtain

$$\begin{aligned} \lim _{k\rightarrow \infty } \frac{f_k}{k} \underline{W}^{(k)}(\pi ) \ge&\lim _{k\rightarrow \infty } \frac{f_k}{k} \sum _{r=1}^R \frac{\mu _r}{\Vert \mu \Vert } \mathbb {E}\mathcal {W}_{r,\kappa }^{(k)}\left( \underline{V}\left( \frac{\mu _r}{\Vert \mu \Vert (k+f_k)}\right) \right) \end{aligned}$$

(61a)

$$\begin{aligned} \ge&\lim _{k\rightarrow \infty } \frac{f_k}{k} \sum _{r=1}^R \frac{\mu _r}{\Vert \mu \Vert } \times \frac{ \lambda _k k \frac{\mu _r}{\Vert \mu \Vert }}{2(k+f_k)} \frac{\sigma _r^2 + \lceil \Vert \mu \Vert \frac{k+f_k}{\mu _r } \rceil \text{ Var }(T_1^{(k)})}{1 - \frac{ \lambda _k k }{\Vert \mu \Vert (k+f_k)} } \end{aligned}$$

(61b)

$$\begin{aligned} \ge&\lim _{k\rightarrow \infty } \sum _{r=1}^R \frac{\mu _r^2}{2 \Vert \mu \Vert } \times f_k \frac{\bar{\sigma }_r^2 }{ f_k + \frac{ o(1/k) k }{\Vert \mu \Vert } } \end{aligned}$$

(61c)

$$\begin{aligned} =&\sum _{r=1}^R \frac{\mu _r^2 \bar{\sigma }_r^2 }{2\Vert \mu \Vert }, \end{aligned}$$

(61d)

where in (61b) we have used the lower bound in (38).