A simple method for computing state probabilities of theM/G/1 andGI/M/1 finite waiting space queues

This paper presents a simple method for computing steady state probabilities at arbitrary and departure epochs of theM/G/1/K queue. The method is recursive and works efficiently for all service time distributions. The only input required for exact evaluation of state probabilities is the Laplace transform of the probability density function of service time. Results for theGI/M/1/K –1 queue have also been obtained from those ofM/G/1/K queue.     

An M/G/1 unreliable retrial queue with two phases of service and Bernoulli admission mechanism

This paper deals with the steady state behaviour of an M^X/G/1 retrial queue with an additional second phase of optional service and unreliable server where breakdowns occur randomly at any instant while serving the customers. Further concept of Bernoulli admission mechanism is also introduced in the model. This model generalizes both the classical M^X/G/1 retrial queue subject to random breakdown and Bernoulli admission mechanism as well as M^X/G/1 queue with second optional service and unreliable server. We carry out an extensive analysis of this model.     

Waiting and sojourn times in a multi-server queue with mixed priorities

We consider a multi-server queue with K priority classes. In this system, customers of the P highest priorities (P<K) can preempt customers with lower priorities, ejecting them from service and sending them back into the queue. Service times are assumed exponential with the same mean for all classes. The Laplace–Stieltjes transforms of waiting times are calculated explicitly and the Laplace–Stieltjes transforms of sojourn times are provided in an implicit form via a system of functional equations. In both cases, moments of any order can be easily calculated. Specifically, we provide formulae for the steady state means and the second moments of waiting times for all priority classes. We also study some approximations of sojourn-time distributions via their moments. In a practical part of our paper, we discuss the use of mixed priorities for different types of Service Level Agreements, including an example based on a real scheduling problem of IT support teams.      

Bottleneck analysis in multiclass closed queueing networks and its application

Asymptotic behavior of queues is studied for large closed multi-class queueing networks consisting of one infinite server station with K classes and M processor sharing (PS) stations. A simple numerical procedure is derived that allows us to identify all bottleneck PS stations. The bottleneck station is defined asymptotically as the station where the number of customers grows proportionally to the total number of customers in the network, as the latter increases simultaneously with service rates at PS stations. For the case when K=M=2, the set of network parameters is identified that corresponds to each of the three possible types of behavior in heavy traffic: both PS stations are bottlenecks, only one PS station is a bottleneck, and a group of two PS stations is a bottleneck while neither PS station forms a bottleneck by itself. In the last case both PS stations are equally loaded by each customer class and their individual queue lengths, normalized by the large parameter, converge to uniformly distributed random variables. These results are directly generalized for arbitrary K=M. Generalizations for K≠M are also indicated. The case of two bottlenecks is illustrated by its application to the problem of dimensioning bandwidth for different data sources in packet-switched communication networks. An engineering rule is provided for determining the link rates such that a service objective on a per-class throughput is satisfied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Optimal Operating Policy for a Random Additional Service Facility

This paper studies the steady state behaviour of a Markovian queue wherein there is a regular service facility serving the units one by one. A search for an additional service facility for the service of a group of units is started when the queue length increases to K (0 < K < L), where L is the maximum waiting space. The search is dropped when the queue length reduces to some tolerable fixed size L - N. The availability time of an additional service facility is a random variable. The model is directed towards finding the optimal operating policy (N,K) for a queueing system with a linear cost structure.     

A Polynomial Factorization Approach for the Discrete Time GIX/>G/1/K Queue

This paper proposes a polynomial factorization approach for queue length distribution of discrete time GI ^X /G/1 and GI _X /G/1/K queues. They are analyzed by using a two-component state model at the arrival and departure instants of customers. The equilibrium state-transition equations of state probabilities are solved by a polynomial factorization method. Finally, the queue length distributions are then obtained as linear combinations of geometric series, whose parameters are evaluated from roots of a characteristic polynomial.     

On the Transient Behavior of the M/M/1 and M/M/1/M Queues

This paper gives a transient analysis of the classic M/M/1 and M/M/1/K queues. Our results are asymptotic as time and queue length become simultaneously large for the infinite capacity queue, and as the system’s storage capacity K becomes large for the finite capacity queue. We give asymptotic expansions for p_n(t), which is the probability that the system contains n customers at time t. We treat several cases of initial conditions and different traffic intensities. The results are based on (i) asymptotic expansion of an exact integral representation for p_n(t) and (ii) applying the ray method to a scaled form of the forward Kolmogorov equation which describes the time evolution of p_n(t).     

Markov-modulatedPH/G/1 queueing systems

We study a PH/G/1 queue in which the arrival process and the service times depend on the state of an underlying Markov chain J(t) on a countable state spaceE. We derive the busy period process, waiting time and idle time of this queueing system. We also study the Markov modulated E_K/G/1 queueing system as a special case.     

Stochastic decomposition in production inventory with service time

We study an (s, S) production inventory system where the processing of inventory requires a positive random amount of time. As a consequence a queue of demands is formed. Demand process is assumed to be Poisson, duration of each service and time required to add an item to the inventory when the production is on, are independent, non-identically distributed exponential random variables. We assume that no customer joins the queue when the inventory level is zero. This assumption leads to an explicit product form solution for the steady state probability vector, using a simple approach. This is despite the fact that there is a strong correlation between the lead-time (the time required to add an item into the inventory) and the number of customers waiting in the system. The technique is: combine the steady state vector of the classical M/M/1 queue and the steady state vector of a production inventory system where the service is instantaneous and no backlogs are allowed. Using a similar technique, the expected length of a production cycle is also obtained explicitly. The optimal values of S and the production switching on level s have been studied for a cost function involving the steady state system performance measures. Since we have obtained explicit expressions for the performance measures, analytic expressions have been derived for calculating the optimal values of S and s.