We introduce several dynamic models of sources of “posts” publishing them on a “newsfeed” in e.g. some social media. All of them have a product-form stationary distribution. We define four metrics relevant to these newsfeeds and compute them in each model.
We study a restricted version of the Erlang-R queueing model that has both server and space constraints, in which the space restriction is administered by two types of admission policies: blocking or holding in a pre-entrant queue. This model, in which customers occupying spaces may be either “needy” (requiring attention from servers) or “content,” captures a wide range of service environments, such as medical units, contact centers, and courts. We study the interplay between the two resource constraints and the reentrant queueing structure. We analyze the connection between staffing, space, and policy decisions and the system performance as a tool for optimizing resource utilization and in determining stability and performance. We develop product form and many-server approximations of performance measures when the blocking policy applies, and then utilize these results to determine approximations for the holding policy as well. We show that system dynamics are captured by a single measure of the fraction time a customer is “needy” in the network, and that returning customers should be explicitly accounted for both in steady-state and transient environments. We show that capacity allocation of both resource constraints (servers and space) should be considered jointly, and derive methodology to determine it. We show that applying a holding policy requires more resources than a blocking policy and demonstrate the application of our policies in two case studies of resource allocation in hospitals.
Semimartingale reflecting Brownian motions (SRBMs) are an integral part of queueing theory. Fluid paths are a standard tool for demonstrating positive recurrence of SRBMs, and their convergence provides both necessary and sufficient conditions for three- and fewer-dimensional SRBMs. Previous examples have been given in the literature for six- and greater-dimensional positive recurrent SRBMs with divergent linear fluid paths. Using predator–prey models, this article provides a family of four-dimensional positive recurrent SRBMs with divergent linear fluid paths, and a family of five-dimensional positive recurrent SRBMs with divergent linear fluid paths and identity covariance matrix. This closes the gap on the number of dimensions under which the positive recurrence of an SRBM does and does not imply convergence of the associated fluid paths.
We study a non-stationary state-dependent Hawkes single-server queueing model with both time-varying arrival and service rates, under the first-come first-served discipline. The arrival process is a Hawkes process with a time-varying baseline rate function, and a self-exciting function that depends on the state of the queue at the arrival times. The service requirements for the jobs are i.i.d. and are met with a common time-varying rate function, so that the realized service times are non-stationary. We establish both the functional law of large numbers (FLLN) and the functional central limit theorem (FCLT) for the joint dynamics of the Hawkes arrival, queueing and workload processes. The limit for the Hawkes arrival and queueing processes in FLLN is given by a set of nonlinear (state-dependent) differential equations with the queueing component having a reflection at zero, and the fluid limit for the workload process is given by a nonlinear functional of the fluid queueing limit involving a time-shifted cumulative service rate function. Since the traffic intensity is both time and state dependent, we analyze the fluid dynamics in the properly defined underloaded and overloaded intervals. The FCLT is proved for the CLT-scaled processes that are centered around the corresponding transient fluid limits, under the underloaded and overloaded intervals. The diffusion limits for the joint Hawkes arrival and queueing processes are given by a set of nonlinear stochastic differential equations with time-varying coefficients; in particular, the queueing limit is a reflected diffusion at zero over the underloaded intervals. The limit for the CLT-scaled workload process takes a complicated form in the overloaded intervals, involving the limits of the time-shifted cumulative service rate function in both fluid and diffusion scales.
Motivated by recent optimality results in matching models, we consider the symmetric W-matching model with three demand classes and two supply classes, and a threshold-based matching policy with finite threshold value T. Under this policy, class-2 demand items are matched to any supply item only when the number of supply items of that class waiting is larger than T. We characterize the stationary distribution of the system for threshold values T=0,1,2 , and show that it admits a product-form solution. To this end, we develop a novel technique for deriving the stationary distribution for these threshold values. Moreover, our analysis shows that using a nonzero threshold can improve system performance. For general threshold values, we develop an approximate analysis and show that the resulting stationary distribution also has a product-form. Using this approximation, we determine the optimal threshold when all demand classes have identical arrival probabilities. Finally, through simulation experiments, we evaluate the accuracy of the approximation and provide evidence that the insights obtained for T=0,1,2 extend to arbitrary threshold values; in particular, a nonzero threshold may minimize the weighted mean holding cost of the system.
