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Motivated by a service platform, we study a two-sided network where heterogeneous demand (customers) and heterogeneous supply (workers) arrive randomly over time to get matched. Customers and workers arrive with a randomly sampled patience time (also known as reneging time in the literature) and are lost if forced to wait longer than that time to be matched. The system dynamics depend on the matching policy, which determines when to match a particular customer class with a particular worker class. Matches between classes use the head-of-line customer and worker from each class. Since customer and worker arrival processes can be very general counting processes, and the reneging times can be sampled from any finite mean distribution that is absolutely continuous, the state descriptor must track the age-in-system for every customer and worker waiting in order to be Markovian, as well as the time elapsed since the last arrival for every class. We develop a measure-valued fluid model that approximates the evolution of the discrete-event stochastic matching model and prove its solution is unique under a fixed matching policy. For a sequence of matching models, we establish a tightness result for the associated sequence of fluid-scaled state descriptors and show that any distributional limit point is a fluid model solution almost surely. When arrival rates are constant, we characterize the invariant states of the fluid model solution and show convergence to these invariant states as time becomes large. Finally, again when arrival rates are constant, we establish another tightness result for the sequence of fluid-scaled state descriptors distributed according to a stationary distribution and show that any subsequence converges to an invariant state. As a consequence, the fluid and time limits can be interchanged, which justifies regarding invariant states as first order approximations to stationary distributions.
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Marked Point Processes in Discrete Time
We develop a general framework for stationary marked point processes in discrete time. We start with a careful analysis of the sample paths. Our initial representation is a sequence {(tj,kj) :j∈Z} of times tj∈Z and marks kj∈K, with batch arrivals (i.e.,tj=tj+1) allowed. We also define alternative interarrival time and sequence representations and show that the three different representations are topologically equivalent. Then, we develop discrete analogs of the familiar stationary stochastic constructs in continuous time: time-stationary and point-stationary random marked point processes, Palm distributions, inversion formulas and Campbell’s theorem with an application to the derivation of a periodic-stationary Little’s law. Along the way,we provide examples to illustrate interesting features of the discrete-time theory.
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- Award ID(s):
- 1634133
- PAR ID:
- 10109562
- Date Published:
- Journal Name:
- Queueing systems
- Volume:
- 92
- ISSN:
- 0257-0130
- Page Range / eLocation ID:
- 47-81
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/doi/10.1007/s11134-019-09612-3
