# Product Forms for FCFS Queueing Models with Arbitrary Server-Job Compatibilities: An Overview

@article{Gardner2020ProductFF, title={Product Forms for FCFS Queueing Models with Arbitrary Server-Job Compatibilities: An Overview}, author={Kristen Gardner and Rhonda Righter}, journal={Queueing Syst. Theory Appl.}, year={2020}, volume={96}, pages={3-51} }

In recent years, a number of models involving different compatibilities between jobs and servers in queueing systems, or between agents and resources in matching systems, have been studied, and, under Markov assumptions and appropriate stability conditions, the stationary distributions have been shown to have product forms. We survey these results and show how, under an appropriate detailed description of the state, many existing product-form results are corollaries of similar results for the…

## 27 Citations

### Matching queues with reneging: a product form solution

- MathematicsQueueing Syst. Theory Appl.
- 2020

A parallel matching queue with reneging is studied and explicit product forms of the steady state distributions of this system are derived by identifying a partial balance condition.

### Pass-and-swap queues

- MathematicsQueueing Syst. Theory Appl.
- 2021

Pass-and-swap is introduced and it is demonstrated that closed networks of P&S queues can be applied to describe the dynamics of new and existing load-distribution and scheduling algorithms in machine pools.

### Directed FCFS infinite bipartite matching

- MathematicsQueueing Syst. Theory Appl.
- 2020

This model may be used for two sided queueing applications such as ride sharing, web purchases, organ transplants, and for abandon at completion redundancy parallel service queues.

### Stochastic non-bipartite matching models and order-independent loss queues

- Mathematics, Computer ScienceStochastic Models
- 2021

This paper proves that the Markov process associated with this model has the same transition diagram as in a product-form queueing model called an order-independent loss queue, which allows to adapt existing results on order- independent (loss) queues to stochastic matching models and derive closed-form expressions for several performance metrics that can be implemented using dynamic programming.

### Redundancy Scheduling with Locally Stable Compatibility Graphs.

- Computer Science
- 2020

This work considers a scenario where jobs of various types are replicated to different subsets of servers as prescribed by a general compatibility graph, and exploits a product-form stationary distribution and weak local stability conditions to establish a state space collapse in heavy traffic.

### A Survey of Stability Results for Redundancy Systems

- Computer ScienceModern Trends in Controlled Stochastic Processes:
- 2021

A survey of recent results concerning the stability arguably the first benchmark of performance of systems with cancel-oncompletion redundancy is given and open questions and conjectures are pointed out.

### Optimal Load Balancing with Locality Constraints

- Computer Science, MathematicsProc. ACM Meas. Anal. Comput. Syst.
- 2020

Under a "well-connected'' graph condition, it is shown that JFSQ and JFIQ are asymptotically optimal in the mean response time when the number of servers goes to infinity, and the well-connectedness condition can be satisfied by a random bipartite graph construction with relatively sparse connectivity.

### Optimal Load Balancing in Bipartite Graphs

- Computer Science, MathematicsArXiv
- 2020

Under a "well-connected" graph condition, it is shown that JFSQ and JFIQ are asymptotically optimal in the mean response time when the number of servers goes to infinity, and the well-connectedness condition can be satisfied by a random bipartite graph construction with relatively sparse connectivity.

### A survey on skill-based routing with applications to service operations management

- Business, Computer ScienceQueueing Syst. Theory Appl.
- 2020

This survey reviews works on skill-based routing in queues and summarizes key insights on routing/scheduling policies developed in the literature, and discusses complications brought by modern service operations management problems, particularly healthcare systems.

### Heavy-Traffic Universality of Redundancy Systems with Assignment Constraints

- Computer ScienceOperations Research
- 2022

Surprisingly, when experiencing high demand, heavy-traffic universality of redundancy systems with assignment constraints can achieve a similar level of resource pooling and performance as a fully flexible system, even subject to quite strict task-server constraints.

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