In open Kelly and Jackson networks, servers are assigned to individual stations, serving customers only where they are assigned. We investigate the performance of modified networks where servers cooperate. A server who would be idle at the assigned station will serve customers at another station, speeding up service there. We assume interchangeable servers: the service rate of a server at a station depends only on the station, not the server. This gives work conservation, which is used in various ways. We investigate three levels of server cooperation, from full cooperation, where all servers are busy when there is work to do anywhere in the network, to one-way cooperation, where a server assigned to one station may assist a server at another, but not the converse. We obtain the same stability conditions for each level and, in a series of examples, obtain substantial performance improvement with server cooperation, even when stations before modification are moderately loaded.

In this paper, we study a large multi-server loss model under the SQ(d) routeing scheme when the service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity, giving rise to a mean field model. The fixed point of the limiting mean field equations (MFEs) was seen to be insensitive to the service time distribution in simulations, but no proof was available. While insensitivity is well known for loss systems, the models, even with state-dependent inputs, belong to the class of linear Markov models. In the context of SQ(d) routeing, the resulting model belongs to the class of nonlinear Markov processes (processes whose generator itself depends on the distribution) for which traditional arguments do not directly apply. Showing insensitivity to the general service time distributions has thus remained an open problem. Obtaining the MFEs in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFEs and then show that the MFEs have a unique fixed point that coincides with the fixed point in the exponential case, thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean field limit.

We introduce a formal limit, which we refer to as a fluid limit, of scaled stochastic models for a cache managed with the least-recently-used algorithm when requests are issued according to general stochastic point processes. We define our fluid limit as a superposition of dependent replications of the original system with smaller item sizes when the number of replications approaches ∞. We derive the average probability that a requested item is not in a cache (average miss probability) in the fluid limit. We show that, when requests follow inhomogeneous Poisson processes, the average miss probability in the fluid limit closely approximates that in the original system. Also, we compare the asymptotic characteristics, as the cache size approaches ∞, of the average miss probability in the fluid limit to those in the original system.

The problem we consider here is a full-information best-choice problem in which n applicants appear sequentially, but each applicant refuses an offer independently of other applicants with known fixed probability 0≤q<1. The objective is to maximize the probability of choosing the best available applicant. Two models are distinguished according to when the availability can be ascertained; the availability is ascertained just after the arrival of the applicant (Model 1), whereas the availability can be ascertained only when an offer is made (Model 2). For Model 1, we can obtain the explicit expressions for the optimal stopping rule and the optimal probability for a given n. A remarkable feature of this model is that, asymptotically (i.e. n→∞), the optimal probability becomes insensitive to q and approaches 0.580 164. The planar Poisson process (PPP) model provides more insight into this phenomenon. For Model 2, the optimal stopping rule depends on the past history in a complicated way and seems to be intractable. We have not solved this model for a finite n but derive, via the PPP approach, a lower bound on the asymptotically optimal probability.

We give a simple and direct treatment of insensitivity in stochastic networks which is quite general and provides probabilistic insight into the phenomenon. In the case of multi-class networks, our results generalise those of Bonald and Proutière (2002), (2003).

We consider an M/G/1 queue that is idle at time 0. The number of customers sampled at an independent exponential time is shown to have the same geometric distribution under the preemptive-resume last-in-first-out and the processor-sharing disciplines. Hence, the marginal distribution of the queue length at any time is identical for both disciplines. We then give a detailed analysis of the time until the first departure for any symmetric queueing discipline. We characterize its distribution and show that it is insensitive to the service discipline. Finally, we study the tail behavior of this distribution.

We obtain a necessary and sufficient condition for the interaction between a service system and an external environment under which the stationary joint distribution of the set of busy channels and the state of the external environment is given by a product-form formula.

A generalized semi-Markov process with reallocation (RGSMP) was introduced to accommodate a large class of stochastic processes which cannot be analyzed by the well-known model of an ordinary generalized semi-Markov process (GSMP). For stationary RGSMP whose initial distribution has a product form, we show that, for a randomly chosen clock of a fixed insensitive type, if the lifetime of this clock is changed to infinity, then the background process is stationary under a certain time change. This implies that the expected time required for the tagged clock to consume a given amount x of resource, called the attained sojourn time, is a linear function of x. Such stationarity and linearity results are known for two special RGSMPs: ordinary GSMP and Kelly's symmetric queue. Our results not only extend them to a general RGSMP but also give more detailed formulas, which allow us to calculate for instance the expected attained sojourn time while the background process is in a given state. Furthermore, we remark that analogous results hold for GSMP with point-process input, in which the lifetimes of clocks of a fixed type form an arbitrary stationary sequence (of not necessarily independent random variables).

Shanthikumar and Sumita (1986) proved that the stationary system queue length distribution just after a departure instant is geometric for GI/GI/1 with LCFS-P/H service discipline and with a constant acceptance probability of an arriving customer, where P denotes preemptive and H is a restarting policy which may depend on the history of preemption. They also got interesting relationships among characteristics. We generalize those results for G/G/1 with an arbitrary restarting LCFS-P and with an arbitrary acceptance policy. Several corollaries are obtained. Fakinos' (1987) and Yamazaki's (1990) expressions for the system queue length distribution are extended. For a Poisson arrival case, we extend the well-known insensitivity for LCFS-P/resume, and discuss the stationary distribution for LCFS-P/repeat.

We study a queueing system with a finite number of input sources. Jobs are individually generated by a source but wait to be served in batches, during which the input of that source is stopped. The service speed of a server depends on the mode of other sources and thus includes interdependencies. The input and service times are allowed to be generally distributed. A classical example is a machine repair system where the machines are subject to shocks causing cumulative damage. A product-form expression is obtained for the steady state joint queue length distribution and shown to be insensitive (i.e. to depend on only mean input and service times). The result is of both practical and theoretical interest as an extension of more standard batch service systems.

Queueing systems are studied with a last-come, first-served queueing discipline and batch arrivals generated by a finite number of non-exponential sources. A closed-form expression is derived for the steady-state queue length distribution. This expression has a scaled geometric form and is insensitive to the input distribution. Moreover, an algorithm for the recursive computation of the normalizing constant and the busy source distribution is presented. The results are of both practical and theoretical interest as an extension of the standard Poisson batch input case.

In [1], [4] and [6] simple and intuitively obvious bounds were suggested for various overflow systems and claimed to be insensitive, that is to be valid also in the non-exponential case. This insensitivity was formalized in [4] and [6] when the service distributions have a monotone failure rate. This technical note provides another proof which for the case of a pure overflow system with one customer class extends the insensitivity to the general case.

Recently a number of authors have considered general procedures for coupling stochastic systems. If the individual components of a system, when considered in isolation, are found to possess the simplifying feature of either reversibility, quasireversibility or partial balance they can be coupled in such a way that the equilibrium analysis of the system is considerably simpler than one might expect in advance. In particular the system usually exhibits a product-form equilibrium distribution and this is often insensitive to the precise specification of the individual components. It is true, however, that certain kinds of components lose their simplifying feature if the specification of the coupling procedure changes. From a practical point of view it is important, therefore, to determine if, and then under what conditions, the revelant feature is preserved.

In this paper we obtain conditions under which partial balance in a component is preserved and these often amount to the requirement that there exists a quantity which is unaffected by the internal workings of the component in question. We give particular attention to the components of a stratified clustering process as these most often suffer from loss of partial balance.

We consider queueing systems where the stationary state probabilities are insensitive with respect to the distribution of certain basic random variables such as service requirements, interarrival times, repair times, etc. The conditional expected sojourn times are stated as Radon–Nikodym densities of the stationary distribution at jump points of the queueing system. The conditions are the given values of such basic random variables for which the insensitivity is valid. We use stationary point processes as our main tool. This means that dependences between certain basic random variables are permitted. Conditional expected real service times, conditional mean response times in closed queueing networks, and similar conditional expected values, are dealt with as special cases.

Generalized semi-Markov schemes were devised to give a versatile general model embracing queueing networks and similar systems of practical importance, and they have proved particularly successful in uniting many disparate results on insensitivity. However, it turns out that, although closed queueing networks are expressible as GSMS, open networks are not, and that the insensitivity results for such networks are not therefore strictly within their scope. In this paper, it is shown that, as one might hope, open networks can be realized as limits of a suitable sequence of closed networks in such a way that the insensitivity properties of the GSMS are transferred to the open network in the limit, and thus that open networks too can, in a sense, be considered to be GSMS. However, it appears from the technical nature of the arguments involved that, despite this close relationship between GSMS and open networks, it may nonetheless be simpler to treat them separately when constructing the proofs of theorems.

For a broad class of stochastic processes, the generalized semi-Markov processes, conditions are known which imply that the steady state distribution of the process, when it exists, depends only on the means, and not the exact shapes, of certain lifetime distributions entering the definition of the process. It is shown in the present paper that this insensitivity extends to certain average and conditional average residence times. Particularly interesting applications can be found in the field of networks of queues.

It is well known that the stationary distribution of the number of busy servers in the Erlang blocking system (M/G/c/c) depends on the service-time distribution only through its mean. This insensitivity property is shared by several other queueing systems. In this paper, we give simple sufficient conditions for determining if this insensitivity property holds for general queueing systems and related stochastic models. The conditions involve determining whether the solution of the stationary Markovian flow equations also solves certain restricted flow equations. The proof that these conditions are sufficient is direct and elementary.

A generalized semi-Markov process with speeds describes the fluctuation, in time, of the state of a certain general system involving, at any given time, one or more living components, whose residual lifetimes are being reduced at state-dependent speeds. Conditions are given for the stationary state distribution, when it exists, to depend only on the means of some of the lifetime distributions, not their exact shapes. This generalizes results of König and Jansen, particularly to the infinite-state case.

The stationary probabilities for certain networks of queues as defined by Kelly [4] were recently shown by Barbour [1] to depend on the service-time distributions involved only through their means. This type of insensitivity has been studied by König and Jansen [5] for a general class of stochastic processes. Kelly's networks yield special cases of such processes. We point this out in the present paper, thus shedding new light on the insensitivity phenomenon observed in these networks and its connection with the phenomenon of local balance. As a consequence of our recent study [8] we also obtain a new insensitivity result for these networks.