On the correlation of sojourn times in open networks of exponential multiserver queues

We consider the joint distribution of successive sojourn times of customers traversing a path in Jackson network. It is shown in a general setting that sojourn times exhibit positive dependence. This especially applies to paths which admit overtaking due to the network topology as well as due to the internal node structure.     

Robustness analysis of generalized Jackson network

Computational Management Science - Queuing networks are a well-established approach to modeling and analysis of complex systems. This paper develops an approach to risk-analysis of queuing network...     

M/M/1 Queueing systems with inventory

We derive stationary distributions of joint queue length and inventory processes in explicit product form for various M/M/1-systems with inventory under continuous review and different inventory management policies, and with lost sales. Demand is Poisson, service times and lead times are exponentially distributed. These distributions are used to calculate performance measures of the respective systems. In case of infinite waiting room the key result is that the limiting distributions of the queue length processes are the same as in the classical M/M/1/∞-system. All authors were supported by DAAD/KBN grant number D/02/32206.     

Throughput limits from the asymptotic profile of cyclic networks with state-dependent service rates

We consider networks where at each node there is a single exponential server with a service rate which is a non-decreasing function of the queue length. The asymptotic profile of a sequence of networks consists of the set of persistent service rates, the limiting customer-to-node ratio, and the limiting service-rate measure. For a sequence of cyclic networks whose asymptotic profile exists, we compute upper and lower bounds for the limit points of the sequence of throughputs as functions of the limiting customer-to-node ratio. We then find conditions under which the limiting throughput exists and is expressible in terms of the asymptotic profile. Under these conditions, we determine the limiting queue-length distributions for persistent service rates. In the absence of these conditions, the limiting throughput need not exist, even for increasing sequences of cyclic networks.      

The cyclic queue and the tandem queue

We consider a closed queueing network, consisting of two FCFS single server queues in series: a queue with general service times and a queue with exponential service times. A fixed number \(N\) of customers cycle through this network. We determine the joint sojourn time distribution of a tagged customer in, first, the general queue and, then, the exponential queue. Subsequently, we indicate how the approach toward this closed system also allows us to study the joint sojourn time distribution of a tagged customer in the equivalent open two-queue system, consisting of FCFS single server queues with general and exponential service times, respectively, in the case that the input process to the first queue is a Poisson process.     

Individual customer’s behaviour in networks with state-dependent arrival rates

We consider Jackson networks with state-dependent arrival and service rates which show product form or nearly product form steady-states and come up with examples of load-dependent admission control. For these networks we prove an arrival theorem for external as well as for internal arrivals. In case of open tandem systems with state-independent service rates we compute the joint distribution of the sojourn times of a customer in the nodes and the distribution of the customer’s end-to-end-delay. This revised version was published online in June 2006 with corrections to the Cover Date.     

The joint distribution of sojourn times for a customer traversing an overtake-free series of queues: The discrete time case

We investigate a cycle of Bernoulli servers in discrete time. We show that the joint distribution of a customer’s sojourn times at the nodes during a round trip is of product form and can explicitly be written down. The results are used to prove independence of sojourn times in an open series of Bernoulli servers. This revised version was published online in June 2006 with corrections to the Cover Date.     

Dependence structure of sojourn times via partition separated ordering

We investigate the joint distribution of successive sojourn times of a customer traversing a path in a Jackson network. It is shown in a general setting that sojourn times for a broad class of such paths exhibit positive dependence. This applies to paths which admit overtaking due to the network topology as well as due to the internal node structure. Our proofs utilize the concept of partition separated orders on multidimensional ordered spaces.     

The cycle time distribution in a cycle of Bernoulli servers in discrete time

In a cycle of Bernoulli servers in discrete time the equilibrium distribution for a customer's round-trip time is shown to be of product-form and is given in explicit formulas. The results are used to obtain the equilibrium flow time distribution for an open tandem of queues.