The stationary Erlang loss model is a classic example of an insensitive queueing system: The steady-state distribution of the number of busy servers depends on the service-time distribution only through its mean. However, when the arrival process is a nonstationary Poisson process, the insensitivity property is lost. We develop a simple effective numerical algorithm for the M t/PH/s/0 model with two service phases and a nonhomogeneous Poisson arrival process, and apply it to show that the time-dependent blocking probability with nonstationary input can be strongly influenced by the service-time distribution beyond the mean. With sinusoidal arrival rates, the peak blocking probability typically increases as the service-time distribution gets less variable. The influence of the service-time distribution, including this seemingly anomalous behavior, can be understood and predicted from the modified-offered-load and stationary-peakedness approximations, which exploit exact results for related infinite-server models. Key Words: nonstationary queues; time-dependent arrival rates; nonhomogeneous Markov chains; transient behavior; Erlang loss model; blocking probability; insensitivity; infinite-server queues; modified-offered-load approximation.

We study uniform acceleration (UA) expansions of finite-state continuous-time Markov chains with time-varying transition rates. The UA expansions can be used to justify, evaluate, and refine the pointwise stationary approximation, which is the steady-state distribution associated with the time-dependent generator at the time of interest. We obtain UA approximations from these UA asymptotic expansions. We derive a time-varying analog to the uniformization representation of transition probabilities for chains with constant transition rates, and apply it to establish asymptotic results related to the UA asymptotic expansion. These asymptotic results can serve as appropriate time-varying analogs to the notions of stationary distributions and limiting distributions. We illustrate the UA approximations by doing a numerical example for the time-varying Erlang loss model. 1 Accepted for publication in the Annals of Applied Probability. AMS 1991 subject classifications. 60J27, 60K30. Keywords...

We relate laws of large numbers and central limit theorems for nonstationary counting processes to corresponding limits for their inverse processes. We apply these results to develop approximations for queues that are unstable in a nonstationary manner. We obtain unstable nonstationary analogs of the queueing relation L = λW and associated central-limit-theorem versions. For modeling and to obtain the first limits, we can construct nonstationary point processes as random time-transformations of familiar point processes, such as renewal processes and stationary point processes. We deduce the asymptotic behavior of the nonstationary point process from the asymptotic behavior of the familiar point process and the time transformation.

We compare the performance of six methods in computing or approximating service levels for nonstationary M/M/s queueing systems: an exact method (a Runge Kutta ordinary differential equation solver), the randomization method, a closure (or surrogate distribution) approximation, a direct infinite server approximation, a modified offered load infinite server approximation, and an effective arrival rate approximation. We used all of the methods to solve the same set of 128 test problems. The randomization method was almost as accurate as the exact method, and used less than half the computational time of the exact method. The closure approximation was less accurate, and in many cases slower, than the randomization method. The two infinite server based approximations and the effective arrival rate approximation had were less accurate but had computation times that were far shorter and less problem-dependent than for the other three methods.

Motivated by interest in making delay announcements to arriving customers who must wait in call centers and related service systems, we study the performance of alternative real-time delay estimators based on recent customer delay experience. The main estimators considered are: (i) the delay of the last customer to enter service (LES), (ii) the delay experienced so far by the customer at the head of the line (HOL), and (iii) the delay experienced by the customer to have arrived most recently among those who have already completed service (RCS). We compare these delay-history estimators to the estimator based on the queue length (QL), which requires knowledge of the mean interval between successive service completions in addition to the queue length. We characterize performance by the mean squared error (MSE). We do analysis and conduct simulations for the standard GI/M/s multi-server queueing model, emphasizing the case of large s. We obtain analytical results for the conditional distribution of the delay given the observed HOL delay. An approximation to its mean value serves as a refined estimator. For all three candidate delay estimators, the MSE relative to the square of the mean is asymptotically negligible in the many-server and classical heavy-traffic limiting regimes.

We study the convergence of finite-capacity open queueing systems to their infinite-capacity counterparts as the capacity increases. Convergence of the transient behavior is easily established in great generality provided that the finite-capacity system can be identified with the infinitecapacity system up to the first time that the capacity is exceeded. Convergence of steady-state distribution is more difficult; it is established here for single-facility models such as GI/GI/c/n with c servers, n - c extra waiting space and the first-come first-served discipline, in which all arrivals finding the waiting room full are lost without affecting future arrivals, via stability properties of generalized semi-Markov processes. 1. Introduction Consider an open queueing system with capacity n. When n is very large, we expect that the standard descriptive stochastic processes, such as the number of customers in the system at time t for t 0, and their limiting steady-state distributions are ve...

We previously introduced and analyzed the Gt/Mt/st +GIt many-server fluid queue with time-varying parameters, intended as an approximation for the corresponding stochastic queueing model when there are many servers and the system experiences periods of overload. In this paper we establish an asymptotic loss of memory (ALOM) property for that fluid model; i.e., we show that there is asymptotic independence from the initial conditions as time t evolves, under regularity conditions. We show that the difference in the performance functions dissipates over time exponentially fast, again under the regularity conditions. We apply ALOM to show that the stationary G/M/s + GI fluid queue converges to steady state and the periodic Gt/Mt/st + GIt fluid queue converges to a periodic steady state as time evolves, for all finite initial conditions.

We develop new improved real-time delay predictors for many-server service systems with a time-varying arrival rate, a time-varying number of servers and customer abandonment. We develop four new predictors, two of which exploit an established deterministic fluid approximation for a many-server queueing model with those features. These delay predictors may be used to make delay announcements. We use computer simulation to show that the proposed predictors outperform previous predictors.

Motivated by interest in making delay announcements in service systems, we study real-time delay estimators in many-server service systems, both with and without customer abandonment. Our main contribution here is to consider the realistic feature of time-varying arrival rates. We focus especially on delay estimators exploiting recent customer delay history. We show that time-varying arrival rates can introduce significant estimation bias in delayhistory-based delay estimators when the system experiences alternating periods of overload and underload. We then introduce refined delay-history estimators that effectively cope with time-varying arrival rates together with non-exponential service-time and abandonment-time distributions, which are often observed in practice. We use computer simulation to verify that our proposed estimators outperform several natural alternatives.

Motivated by interest in making delay announcements in service systems, we study real-time delay estimators in many-server service systems, both with and without customer abandonment. Our main contribution here is to consider the realistic feature of time-varying arrival rates. We focus especially on delay estimators exploiting recent customer delay history. We show that time-varying arrival rates can introduce significant estimation bias in delayhistory-based delay estimators when the system experiences alternating periods of overload and underload. We then introduce refined delay-history estimators that effectively cope with time-varying arrival rates together with non-exponential service-time and abandonment-time distributions, which are often observed in practice. We use computer simulation to verify that our proposed estimators outperform several natural alternatives.