Decision-Epoch Matters: Unveiling its Impact on the Stability of Scheduling with Randomly Varying Connectivity

TL;DR

It is shown that Serve Longest Connected queue is maximum stable in both constrained settings, within the set of policies that select a queue among the connected ones, and a novel theoretical tool termed a test for fluid limits (TFL) that might be of independent interest.

Abstract

A classical queuing theory result states that in a parallel-queue single-server model, the maximum stability region does not depend on the scheduling decision epochs, and in particular is the same for preemptive and non-preemptive systems. We consider here the case in which each of the queues may be connected to the server or not, depending on an exogenous process. In our main result, we show that the maximum stability region now does strongly depend on how the decision epochs are defined. We compare the setting where decisions can be made at any moment in time (the unconstrained setting), to two other settings: decisions are taken either (i) at moments of a departure (non-preemptive scheduling), or (ii) when an exponentially clock rings with rate $γ$. We characterise the maximum stability region for the two constrained configurations, allowing us to observe a reduction compared to the unconstrained configuration. In the non-preemptive setting, the maximum stability region is drastically reduced compared to the unconstrained setting and we conclude that a non-preemptive scheduler cannot take opportunistically advantage (in terms of stability) of the random varying connectivity. Instead, for the $γ$ decision epochs, we observe that the maximum stability region is monotone in the rate of the decision moments $γ$, and that one can be arbitrarily close to the maximum stability region in the unconstrained setting if we choose $γ$ large enough. We further show that Serve Longest Connected (SLC) queue is maximum stable in both constrained settings, within the set of policies that select a queue among the connected ones. From a methodological viewpoint, we introduce a novel theoretical tool termed a ``test for fluid limits'' (TFL) that might be of independent interest. TFL is a simple test that, if satisfied by the fluid limit, allows us to conclude for stability.

Figures (8)

• Figure 1: Stability regions with $K=2$ for Settings I (dashed line), II (dotted line) and III (dash-dotted line, for relatively small and large$\gamma$).
• Figure 2: For $K=2$, we depict $\mathrm{MSR}_{\mathrm{I}}$(left), $\mathrm{MSR}_{\mathrm{II}}$(center) and $\mathrm{MSR}_{\mathrm{III}}$(right).
• Figure 3: This picture represents condition \ref{['eqn:ladrillo']}. The three leading lines are the graphs of the functions $G_i$ for which $i$ attains the maximum in the extremes of the interval under consideration, $[t_1,t_2]$. The other lines are the graphs of the remaining $G_i$'s. The two groups of functions are not allowed to intersect among the hole interval $[t_1,t_2]$.
• Figure 4: Condition \ref{['eqn:cond1']}
• Figure 5: Condition \ref{['eqn:cond2']}

Theorems & Definitions (14)

• Remark 1: Markovian assumptions
• Definition 3.1: class $\mathcal{P}$ of policies
• Theorem 4.1
• Proposition 4.2
• proof
• Corollary 4.3
• Corollary 4.4
• Corollary 4.5
• Corollary 4.6
• Proposition 4.7