Asymptotic analysis and simple approximation of the loss probability of the GI/M/c/K queue

We consider the GI^X/M/c/K queues with partial rejection or total rejection, and find an asymptotic behavior of loss probability of the GI^X/M/c/K queue as K tends to infinity. The asymptotic loss probability is expressed only in terms of the roots of the characteristic equation and the boundary probabilities of the corresponding GI^X/M/c queue. Numerical examples show that the asymptotic loss probability is a quite accurate approximation for the loss probability of the GI^X/M/c/K queue even when the system capacity K is moderate.     

Analysis on queueing systems with synchronous vacations of partial servers

We study an M/M/c queue with server vacations. In this queueing system, d (≤c) of c servers take synchronous vacations when these d servers become idle at a service completion instant. This type of queueing model captures the major characteristics of a stochastic service system which processes both random arriving jobs and constantly available jobs. In this paper, the multi-server vacation queueing system has been analyzed as a quasi-birth and death process. Using matrix geometric method, we obtain the stationary distributions of queue length and waiting time and demonstrate the conditional stochastic decomposition property of the queue length and waiting time in this system.     

On the overflow processes from the PH1 + PH2/M/S/K queue with two independent PH-renewal inputs

This paper presents an analysis of overflow processes from a PH₁ + PH₂/M/S/K queue having two independent phase type renewal input streams. Both the superposed overflow process and individual overflow processes for the PH₁- and PH₂-streams are analyzed using first passage time distributions for the number of customers in the system. Each overflow process is characterized as a Markov renewal process. The nth moment of the number of customers in an infinite server group to which these overflows have been offered is derived using a theory for the MR/M/∞ queue with a Markov renewal input. The numerical examples for means and variance-to-mean ratios (peakednesses) of the individual overflow streams are given for an H₂ + H₂/M/S/S queue with interrupted Poisson inputs, which is a vital model for telephone network planning. In addition, overflow traffic characteristics are discussed by using these examples.     

Sojourn time distribution in the M/M/1 queue with discriminatory processor-sharing

In this paper, we consider a queue with multiple K job classes, Poisson arrivals, exponentially distributed required service times in which a single processor serves according to the DPS discipline. More precisely, if there are n_i class i jobs in the system, i=1,…,K, each class j job receives a fraction _j/∑_i=1^K_in_i of the processor capacity. For this queue, we obtain a system of equations for joint transforms of the sojourn time and the number of jobs. Using this system of equations we find the moments of the sojourn time as a solution of linear simultaneous equations, which solves an open problem.     

A M/D/1 vacation queue model for a signalized intersection

We consider a queueing system that arises in the modeling of isolated signalized intersections in a urban transportation network. In this system, the server alternates in two states, attended or removed, in respect to the queue, while in each state, the server will spend a constant time period with different value. It is assumed that the server is able to disperse up to r(r≥1) customers during a constant service cycle. The evolution of this queueing system can be characterized by a Markov chain embedded at equally spaced time epochs along the time axis. Transition matrix of this Markov chain is of the M/G/1 type introduced by Neuts so that matrix analytical method can be applied to obtain the necessary and sufficient criterion for ergodicity of this Markov chain as well as to compute its stationary distribution. Furthermore, the queue length and waiting time distributions with other performance measures are also given in this paper.     

Analysis of a finite buffer queue with different scheduling and push-out schemes

A queueing system M₁, M₂/G₁, G₂/1/N with different scheduling and push-out scheme is analyzed in this paper. This work is motivated by the study of the performance of an output link of ATM switches with traffic of two classes with different priorities. However, the queueing model developed in this paper is more general than that of the output link of ATM switches with two-class priority traffic. General service time distributions are allowed for classes 1 and 2 and a general service discipline function, ₁(i, j), is introduced where ₁(i, j) is the probability that a class 1 packet will be served, given that there are i class 1 and j class 2 packets waiting for service. An exact solution is obtained for the loss probabilities for classes 1 and 2, the queue length distribution and the mean waiting time for class 1. The queue length distribution and the mean waiting time for class 2 are calculated approximately. It is shown that the approximation is an upper bound and the error due to the approximation is very small when the loss probability of class 2 is small (e.g., less than 0.01).     

Impact of Network Dynamics on User's Video Quality: Analytical Framework and QoS Provision

We develop an analytical framework to investigate the impacts of network dynamics on the user perceived video quality. Our investigation stands from the end user's perspective by analyzing the receiver playout buffer. In specific, we model the playback buffer at the receiver by a G/G/1/? and G/G/1/N queue, respectively, with arbitrary patterns of packet arrival and playback. We then examine the transient queue length of the buffer using the diffusion approximation. We obtain the closed-form expressions of the video quality in terms of the start-up delay, fluency of video playback and packet loss, and represent them by the network statistics, i.e., the average network throughput and delay jitter. Based on the analytical framework, we propose adaptive playout buffer management schemes to optimally manage the threshold of video playback towards the maximal user utility, according to different quality-of-service requirements of end users. The proposed framework is validated by extensive simulations.     

Algorithms for generalized vertex-rankings of partial k-trees

A c-vertex-ranking of a graph G for a positive integer c is a labeling of the vertices of G with integers such that, for any label i, deletion of all vertices with labels >i leaves connected components, each having at most c vertices with label i. A c-vertex-ranking is optimal if the number of labels used is as small as possible. We present sequential and parallel algorithms to find an optimal c-vertex-ranking of a partial k-tree, that is, a graph of treewidth bounded by a fixed integer k. The sequential algorithm takes polynomial-time for any positive integer c. The parallel algorithm takes O(log n) parallel time using a polynomial number of processors on the common CRCW PRAM, where n is the number of vertices in G.     

Efficient algorithms for shortest distance queries on special classes of polygons

The problem of finding a rectilinear minimum bend path (RMBP) between two designated points inside a rectilinear polygon has applications in robotics and motion planning. In this paper, we present efficient algorithms to solve the query version of the RMBP problem for special classes of rectilinear polygons given their visibility graphs. Specifically, we show that given an unweighted graph G = (V, E), with ¦V¦ = N and ¦E¦ = M, algorithms to preprocess G in linear space and time such that the shortest distance queries — queries asking for the distance between any pair of nodes in the graph — can be answered in constant time and space are presented in this paper. For the case of a chordal graph G, our algorithms give a distance which is at most one away from the actual shortest distance. When G is a K-chordal graph, our algorithm produces an exact shortest distance in O(K) time. We also present a non-trivial parallel implementation of the sequential preprocessing algorithm for the CREW-PRAM model which runs in O(log² N) time using O(N + M) processors. After the preprocessing, we can answer the queries in constant time using a single processor.     

Analysis of the resequencing buffer in a homogeneous M/M/2 queue

The resequencing problem is encountered in many practical information systems such as distributed database and communication networks. In these systems customers, such as messages in a computer network, have to be delivered to users in their original order. Therefore, those customers which become out of order due to the randomness of the system are forced to wait in a resequencing buffer so that their delivered order can be guaranteed. The previous work on the resequencing problem mainly concentrated on the delay aspect. From both theoretical and practical viewpoints, however, the queue length characteristics of the resequencing buffer are also significant. We consider the queue length distribution of the resequencing buffer fed by a homogeneous M/M/2 queue. The exact analysis is carried out for the probability mass functions of the queue length in equilibrium and the maximal occupancy which corresponds to the queue length just before the departure instants of customers from the resequencing buffer.     

Randomized control of T-policy for an M/G/1 system

We study an M/G/1 queueing system with a server that can be switched on and off. The server can take a vacation time T after the system becomes empty. In this paper, we investigate a randomized policy to control a server with which, when the system is empty, the server can be switched off with probability p and take a vacation or left on with probability (1 − p) and continue to serve the arriving customers. For this system, we consider the operating cost and the holding cost where the operating cost consists of the system running and switching costs (start up and shut down costs). We describe the structure and characteristics of this policy and solve a constrained problem to minimize the average operating cost per unit time under the constraint for the holding cost per unit time.     

Computing steady state probabilities in λ(n)/G/1/K queue

In this paper a recursive method is developed to obtain the steady state probability distribution of the number in system at arbitrary and departure time epochs of a single server state-dependent arrival rate queue λ(n)/G/1/K in which the arrival process is Markovian with arrival rates λ(n) which depend on the number of customers n in the system and general service time distribution. It is assumed that there exists an integer K such that λ(n) > 0 for all 0 n < K and λ(n) = 0 for all n K. Numerical results have been presented for many queueing models by suitably defining the function λ(n). These include machine interference model, queues with balking, queues with finite waiting space and machine interference model with finite waiting space. These models have wide application in computer/communication networks.     

Algorithms of discretization of algebraic spatial curves on homogeneous cubical grids

In this paper new methods of discretization (integer approximation) of algebraic spatial curves in the form of intersecting surfaces P(x, y, z) = 0 and Q(x, y, z) = 0 are analyzed.

The use of homogeneous cubical grids G(h³) to discretize a curve is the essence of the method. Two new algorithms of discretization (on 6-connected grid G_6c(h³) and 26-connected grid G₂₆(h³)) are presented based on the method above. Implementation of the algorithms for algebraic spatial curves is suggested. The elaborated algorithms are adjusted for application in computer graphics and numerical control of machine tools.     

Uniform controllable sets of left-invariant vector fields on compact Lie groups

A set of vector fields on a differentiable manifold M is said to be uniformly completely controllable (u.c.c.) if there exists a nonnegative integer N such that evert pair (p, q) of point of M can be joined by a trajectory, or positive orbit, of which involves at most N switches.

In this article we show that if M is a Lie group G and a set of left-invariant vector fields on G, N must be greater than or equal to dim(G)-1. We also construct sets of vector fields which are uniformly completely controllable in dim(G)-1 switches when G is the Lie group of any compact real form of g and g runs over all classical simple Lie algebras over .     

Complex random sample scheduling and its application to an N/M/F/Fmax problem

“Complex Random Sample Scheduling(CRSS)” was proposed in this paper as an efficient heuristic method for solving any permutation scheduling problems. To show the effectiveness of the proposed CRSS, it was applied to an N-job, M-machine, permutation flowshop scheduling problem to minimize makespan, N/M/F/F_max. Numerical experiments made it clear that the proposed CRSS provides a schedule very close to the near-optimal schedule obtained by the existing promising heuristic methods such as taboo search and simulated annealing, within less computation time than these heuristic methods.     

Moments of conjugacy classes of binary words

For each nonempty binary word w=c₁c₂c_q, where c_i{0,1}, the nonnegative integer ∑_i=1^q (q+1−i)c_i is called the moment of w and is denoted by M(w). Let [w] denote the conjugacy class of w. Define M([w])={M(u): u[w]}, N(w)={M(u)−M(w): u[w]} and δ(w)=max{M(u)−M(v): u,v[w]}. Using these objects, we obtain equivalent conditions for a binary word to be an -word (respectively, a power of an -word). For instance, we prove that the following statements are equivalent for any binary word w with |w|2: (a) w is an -word, (b) δ(w)=|w|−1, (c) w is a cyclic balanced primitive word, (d) M([w]) is a set of |w| consecutive positive integers, (e) N(w) is a set of |w| consecutive integers and 0N(w), (f) w is primitive and [w]St.     

Paired-domination in inflated graphs

The inflation G_I of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique K_d. A set S of vertices in a graph G is a paired dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired domination number γ_p(G) is the minimum cardinality of a paired dominating set of G. In this paper, we show that if a graph G has a minimum degree δ(G)2, then n(G)γ_p(G_I)4m(G)/[δ(G)+1], and the equality γ_p(G_I) = n(G) holds if and only if G has a perfect matching. In addition, we present a linear time algorithm to compute a minimum paired-dominating set for an inflation tree.     

Heaps and heapsort on secondary storage

A heap structure designed for secondary storage is suggested that tries to make the best use of the available buffer space in primary memory. The heap is a complete multi-way tree, with multi-page blocks of records as nodes, satisfying a generalized heap property. A special feature of the tree is that the nodes may be partially filled, as in B-trees. The structure is complemented with priority-queue operations insert and delete-max. When handling a sequence of S operations, the number of page transfers performed is shown to be O(∑_{i = 1}^S(1/P) log_(M/P)(N_i/P)), where P denotes the number of records fitting into a page, M the capacity of the buffer space in records, and N_i, the number of records in the heap prior to the ith operation (assuming P 1 and S> M c · P, where c is a small positive constant). The number of comparisons required when handling the sequence is O(∑_{i = 1}^S log₂ N_i). Using the suggested data structure we obtain an optimal external heapsort that performs O((N/P) log_(M/P)(N/P)) page transfers and O(N log₂ N) comparisons in the worst case when sorting N records.     

Entropy maximised queueing networks with blocking and multiple job classes

The method of entropy maximisation (MEM) is applied in a state space partitioning mode for the approximation of the joint stationary queue length distribution of an M/M/1/N queue with finite capacity, N( > 1), multiple and distinct classes of jobs, R( > 1), under a complete buffer sharing scheme and mixed service disciplines drawn from the first-come-first-served (FCFS), last-come-first-served with (LCFS-PR) or without (LCFS-NPR) preemption and processor sharing (PS) rules. The marginal and aggregate maximum entropy (ME) queue length distributions and the associated blocking probabilities per class are also determined. These ME results in conjunction with the first moments of the effective flows are used, as building blocks, in order to establish a new product-form approximation for arbitrary exponential open queueing networks with multiple classes of jobs under repetitive-service (RS) blocking with random destination (RD). It is verified that the ME approximation reduces to the exact truncated solution of open multi-class reversible queueing networks. Numerical experiments demonstrate a good accuracy level of ME statistics in relation to simulation. Moreover, recent extentions of MEM for arbitrary GE-type queueing networks with RS-RD blocking and multiple classes of jobs are presented.