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Abstract Continuous time Markov chains are commonly used as models for the stochastic behavior of chemical reaction networks. More precisely, these Stochastic Chemical Reaction Networks (SCRNs) are frequently used to gain a mechanistic understanding of how chemical reaction rate parameters impact the stochastic behavior of these systems. One property of interest is mean first passage times (MFPTs) between states. However, deriving explicit formulas for MFPTs can be highly complex. In order to address this problem, we first introduce the concept of$$coclique\, level\, structure$$and develop theorems to determine whether certain SCRNs have this feature by studying associated graphs. Additionally, we develop an algorithm to identify, under specific assumptions, all possible coclique level structures associated with a given SCRN. Finally, we demonstrate how the presence of such a structure in a SCRN allows us to derive closed form formulas for both upper and lower bounds for the MFPTs. Our methods can be applied to SCRNs taking values in a generic finite state space and can also be applied to models with non-mass-action kinetics. We illustrate our results with examples from the biological areas of epigenetics, neurobiology and ecology. 

Abstract In this paper, we consider a multi-server, multiclass queue with reneging operating under the random order of service discipline. Interarrival times, service times, and patience times are assumed to be generally distributed. Under mild conditions, we establish a fluid limit theorem for a measure-valued process that keeps track of the remaining patience time for each job in the queue, when the number of servers and classes is held fixed. We prove uniqueness for fluid model solutions in all but one case. We characterize the unique invariant state for the fluid model and prove that fluid model solutions converge to the invariant state as time goes to infinity, uniformly for suitable initial conditions.