Optimal Partition for Multi-Type Queueing System

TL;DR

This work addresses optimal server partitioning and customer-type routing in a divisible-capacity, multi-queue FCFS system with heterogeneous types. It develops a moment-space geometric framework that yields tractable structure for the optimal assignment, with an $O(n^2)$ configuration enumeration for the two-queue case and scalable extensions to $k$ queues through configuration sets in the mean-variance space. The paper shows that partitioning can yield arbitrarily large improvements in waiting (and sojourn) time and that naive two-segment rules by service rate are often suboptimal, while under exponential service and moment-regularity conditions the optimum has a deterministic, two-block structure. It also provides extensions to non-identical waiting costs and sojourn-time objectives, plus practical considerations for type selection and clustering, culminating in efficient algorithms for both two- and multi-queue settings and broad applicability to real-world queueing systems.

Abstract

We study an optimal server partition and customer assignment problem for an uncapacitated FCFS queueing system with heterogeneous types of customers. Each type of customers is associated with a Poisson arrival, a certain service time distribution, and a unit waiting cost. The goal is to minimize the expected total waiting cost by partitioning the server into sub-queues, each with a smaller service capacity, and routing customer types probabilistically. First, we show that by properly partitioning the queue, it is possible to reduce the expected waiting costs by an arbitrarily large ratio. Then, we show that for any given server partition, the optimal customer assignment admits a certain geometric structure, enabling an efficient algorithm to find the optimal assignment. Such an optimal structure also applies when minimizing the expected sojourn time. Finally, we consider the joint partition-assignment optimization problem. The customer assignment under the optimal server partition admits a stronger structure. Specifically, if the first two moments of the service time distributions satisfy certain properties, it is optimal to deterministically assign customer types with consecutive service rates to the same sub-queue. This structure allows for more efficient algorithms. Overall, the common rule of thumb to partition customers into continuous segments ranked by service rates could be suboptimal, and our work is the first to comprehensively study the queue partition problem based on customer types.

Figures (7)

• Figure 1: There are 5 types of customers, and the induced partition by $l(\cdot)$ is $U_{\mathcal{I}, l} = \{1, 5\}, E_{\mathcal{I}, l} = \{2, 4\}, L_{\mathcal{I}, l} = \{3\}$.
• Figure 2: Recover optimal assignment from $(z^{ * }, y^{ * }) = (4, 3)$ with $E = \{1, 2\}$ where $\mu_1 > \mu_2$.
• Figure 3: Objective function value under the assignment in the form of $(0, x_2, 1), (1, x_2, 0)$ and $(1, x_2, 1)$.
• Figure 4: There are 9 types of customers and 3 queues. The induced sets are $T_{\mathcal{P}, 1} = \{9\}, E_{\mathcal{P}, 1} = \{1, 2, 3, 4\}, T_{\mathcal{P}, 2} = \{7\}, E_{\mathcal{P}, 2} = \{1, 2, 5, 6\}, T_{\mathcal{P}, 3} = \{8\}, E_{\mathcal{P}, 3} = \{3, 4, 5, 6\}$.
• Figure 5: There are 5 blocks in $X_1$ and 6 blocks in $X_2$, where blocks are marked as the shaded area.

Theorems & Definitions (20)

• Proposition 1
• Example 1
• Lemma 1
• Theorem 1
• Example 2
• Theorem 2
• Example 3
• Theorem 3
• Corollary 1
• Theorem 4