CORS / Optimization Days
HEC Montréal, May 2931, 2023
CORSJOPT2023
HEC Montreal, 29 — 31 May 2023
QM Queueing Models
May 31, 2023 10:30 AM – 12:10 PM
Location: Xerox Canada (yellow)
Chaired by Elham Heydarigharaei
4 Presentations

10:30 AM  10:55 AM
Estimating the Average Waiting Time in a System with TimeVarying Arrival Rates of the Third Degree
TimeVarying Little’s Law (TVLL) can be regarded as part of the theory of Infinite Servers (IS) models, for the abstract system can be considered as a general IS model if waiting time is considered as service time. Moreover, the timevarying arrival rate does not affect the waiting time distribution, when there are adequate timevarying servers in the system. In this study, we estimate the average number of entities in the system over a subinterval and the arrival rate function, and apply TVLL combined with timevarying staffing to estimate the unknown mean wait times. When the arrival rate function is approximated by a linear (quadratic) function, the average waiting time satisfies a quadratic (cubic) equation. The estimation of average waiting time based on TVLL is a positive real root of the average waiting time equation.
If, the arrival rate function is neither approximately linear nor approximately quadratic, it must be approximated by a polynomial function of higher degree. In this study, we investigate systems with arrival rate function of degree 3, and
find the estimation of average waiting time which is the root of a polynomial of degree 4. 
10:55 AM  11:20 AM
How Meaningful are Equilibrium Solutions in Discrete Event Systems
In queueing systems and other discrete event systems it is often simpler to find equilibrium solutions than transient solutions. This begs the question if these solutions form a sound basis for making decisions in view of the fact that queuing systems operate in a constantly changing environment, and no equilibrium is ever reached. We therefore investigate the error that arises by using equilibrium rather than transient solutions. In order to do that, we investigate the speed at which equilibria are approached. As it turns out, this depends on many factors. Aside from the system structure, and the parameter values, the speed of convergence also depends on the particular measure to be obtained. In particular, in queueing systems, the server usage reaches an equilibrium much faster than the expected queue length. Both theoretical considerations and numerical results will be given.

11:20 AM  11:45 AM
Potty Parity: Process Flexibility via Unisex Restroom
We study the problem of inequitable access to public restrooms by women and the LGBTQ+ community. Individuals enter a restroom based on their gender identity and the expected (or observed) wait time. We consider two measures of potty parity: first, the conventional waittime parity, and second, our proposed utility parity, which encompasses both wait time and gender identity to estimate users' utility for using a restroom. We show the benefits of unisex restrooms analytically and from various angles: (a) reducing the wait time for the women's restroom; (b) enhancing the potty parity of wait times and users' utility; (c) increasing users' feelings of safety; and (d) shrinking the waittime disparity when arrival rates fluctuate. Moreover, we provide insights into both renovating existing buildings and designing restrooms from scratch. In particular, we show the following: (i) The process flexibility of having a oneunit unisex restroom, either by converting a unit of the men's restroom or building an additional one, goes a long way toward improving wait time or user utility, and reducing their disparities. (ii) Building the women's room and the unisex restroom next to each other (such that users can jockey lines) improves potty parity. (iii) Even though an allunisex restroom leads to complete parity of wait times, surprisingly, it does not improve utility potty parity, but reverses the ranking of users' utility in the population. (iv) Providing an allunisex room plus urinal(s) can increase efficiency still more. Lastly, we also provide a numerical study with empirically calibrated parameters to show the magnitude of the impact of unisex rooms in a stadium.

11:45 AM  12:10 PM
Crowd Sensing Congestion in Queuing
In crowdsensed platforms, such as those used for monitoring environmental pollution or traffic flow, a subset of users are equipped to share their observations on a phenomenon of common interest, while all users have access to the information provided by these equipped users. Equipped users can improve the accuracy of available information gathered from swarm intelligence. Employing usergenerated data eliminates the need for largescale data collection infrastructure, benefiting everyone by enabling informed decisionmaking without contractual sharing obligations. To investigate the properties of crowdsensing data gathering, we employ an M/M/1/K queue to analyze measures such as the expected queue length, throughput, and social welfare under different congestion levels and equipped user ratios. The results indicate that the expected queue length decreases with the equipped user ratio, irrespective of the congestion level. However, changes in throughput and social welfare depend on the congestion level. We observe that an equipped user ratio of less than one maximizes social welfare in congested systems.
Keywords: Crowd sensing, Usergenerated data, Observable queue, Unobservable queue