Stability of Global LIFO Networks

It is proved that a general open multi-class queueing network with the Global preempt-resume Last-In-First-Out (G-LIFO) multi-channel discipline is stable under the standard sub-criticality condition j < rj . Here rj is the number of service channels and j = X (i;k)2Gj ivik the `nominal' load in node j; i is the exogenous rate of arrival of customers of class i and vik the mean service time of the class i customers at the kth node on the route. Gj is the set of types (i; k) of the customers served in node j. This result contrasts to examples of multi-class FIFO queueing networks where the nominal sub-criticality condition does not guarantee stability. 1 The G-LIFO service discipline Consider an open queueing network with several customer classes. The class of a customer determines its route through the network and the distribution of the service time in each node on the route. At the time of arrival from outside a customer enters the rst node of its route. After nishing service in a given node, a customer instantly enters the next node of its route; if it was the last node of the route, the customer leaves the network. The queueing discipline adopted in this paper is multi-channel Global preempt-resume Last-In-First-Out (G-LIFO). This means that customers in each node are served in the inverse order of their exogenous arrival times (i.e., times of arrival in the network); a new customer interrupts the current service if the customer under service is `older'. More precisely, suppose that a new customer, say A, with an exogenous arrival time t, enters a given node (from outside or after being served in the previous node of its route). If there is an idle channel in the node, the newcomer takes it. Otherwise, two cases can occur: a) all customers under service are `younger' than A, i.e., have exogenous times t, and b) the `oldest' customer under service has an exogenous time t < t; call it A. In case a) customer A waits until the time when (i) one of the channels completes service and (ii) all other waiting customers (regardless of whether their service was interrupted or not) have their exogenous times < t. Then A occupies the available channel. In case b) A interrupts the service of customer A and takes over the corresponding channel. The interrupted service is then resumed at the rst time when a channel in the node completes service and all other waiting customers have their exogenous times < t. All ties are broken at random (in the case of continuously distributed random variables they occur with probability 0). So, between points of exogenous arrival and end of service the network functions `smoothly': customers under service diminish their residual service times at rate one, while the age (i.e., the time lapsed from the time of exogenous arrival) of all customers present in the network (i.e., served or waiting in the network nodes) increases at rate one. We show that if the nominal load at each node in the network is strictly less than the number of channels then the network is stable. By stability we mean ergodicity (more precisely, positive Harris recurrence) of the underlying stochastic process. This fact is in contrast with the well-known results (see [2], [3]) showing that in the case of the FIFO (First-In-First-Out) discipline, the nominal sub-criticality condition is not enough for stability of a multi-class network. It has to be said that the G-LIFO discipline has its advantages (it is simple to implement, distributes the workload evenly between the nodes of the route) and disadvantages (creates a backlog of interrupted work). It is easy to check that if the nominal sub-criticality condition is reversed in at least one node j 2 J (in the sense that the nominal load is strictly greater than the number of channels in the node) then the network is unstable. The proofs given in this paper are based on the method of the uid limit [12, 5, 4, 13, 6],