Timothy Brown

Participant: PROMISE AGEP Research Symposium

Timothy Brown
: Department of Mathematics & Statistics
Institution: University of Maryland, Baltimore County (UMBC)



Simulation of Fluid Limits of Many-Server Retrial Queues with Non-Persistent Customers

This work considers the limit of fluid model for many-server retrial queues with impatient customers and algorithms to compute quantities associated with the fluid model. A newly arrived customer attempts to obtain service immediately upon arrival and joins a retrial orbit with probability p∈[0,1] if all servers are busy, and re-attempts to obtain service after a random amount of time until it gets service. The dynamics of the system is represented in terms of a family of measure-valued processes that keep track of the amounts of time that each customer being served has been in service and the waiting times of customers in the retrial orbit since their previous attempts to obtain service. Under some mild assumptions, as both the arrival rate and the number of servers go to infinity, a law of large numbers (or fluid) limit is established for this family of processes with the aid of the one-dimensional Skorokhod map and a contraction map. The limit is shown to be the unique solution to the so-called (extended) fluid model equations.




Let D be a set with discrete topology P(D). Let βD be the set of all ultrafilters on D and let e: D → βD  be e(x) is the principal ultrafilter generated by x ∈  D. We define a topology τ on βD and prove the following theorem.

Theorem: (e , βD) is a Stone-Cech compactification of D.

The Stone-Cech compactification is important because it allows us to produce new results. For instance, suppose we are given any set Y which is not compact Hausdorff space. We can look for a compact Hausdorff space which is very similar to Y. That is, there is a homeomorphism h going from Y into a subspace h(Y) on the compact Hausdorff T. Now we can use results in T and translate it back to Y.


Tim Brown earned his Bachelor of Science degree in Mathematics from Morgan State University. In fall 2014 he joined the graduate program in Applied Mathematics at UMBC. Mr. Brown research interests are queuing theory, applied probability,stochatic modelling,big data, and performance modelling of telecommunications and health care services.



We have seen that as a system gets congested, the service delay in the system increases. A good understanding of the relationships between congestion and delay is essential for designing effective congestion control algorithms. Queuing theory provides all the tools needed for this analysis. A  queue can be studied in terms of the source of each queued item, how frequently items arrive on the queue, how long they can or should wait, whether some items should jump ahead in the queue, how multiple queues might be formed and managed, and the rules by which items are queued and not queued. Providing too much service involves excessive costs. And not providing enough service capacity causes the waiting line to become excessively long. Queuing theory is important because it examines the leasing of shared but limited resources and strategic decision making. My current research provides a fluid approximations of the system dynamics in the many server heavy traffic region where the external arrival rate to the server pool and the number of servers go to infinity together in an appropriate manner, while the service and inter-attempt time distributions are fixed.


  1. Distributional Rank Aggregation, and Axiomatic Analysis, University of Maryland Baltimore County (2015)
  2. Google’s Page Rank Algorithm, Graduate Semianr,University of Maryland Baltimore County (2015)
  3. The Stone Cech Compafaction of a Discrete Space, 2016 Promise Agep Research Symposium.

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