Analysis of the adaptive MMAP[K]/PH[K]/1 queue: A multi-type queue with adaptive arrivals and general impatience

In this paper we introduce the adaptive MMAP[K] arrival process and analyze the adaptive MMAP[K]/PH[K]/1 queue. In such a queueing system, customers of K different types with Markovian inter-arrival times and possibly correlated customer types, are fed to a single server queue that makes use of r thresholds. Service times are phase-type and depend on the type of customer in service. Type k customers are accepted with some probability a_i,k if the current workload is between threshold i − 1 and i. The manner in which the arrival process changes its state after generating a type k customer also depends on whether the customer is accepted or rejected.     

On optimal right-of-way policies at a single-server station when insertion of idle times is permitted

A general stream of n types of customers arrives at a Single Server station where service is non-preemptive, the server may undergo Poisson breakdowns and insertion of idle times is allowed. If ξ(k) and c(k) are, respectively, the expected service time and sojourn cost per unit time of a type k customer (1?k?n), call k “V.I.P.” type if ξ(k)/c(k) = min_1?i?n[ξ(i)/sbc(i)].We show that any right-of-way service policy can be improved by a policy that grants V.I.P. customers priority over all others, and never inserts idle time when a V.I.P. customer is present.We further show that if the arrival stream is Poisson, the so-called “cμ” priority rule (applied with no delays) is optimal in the class of all service policies, and not just among those of a priority nature.     

The shorter queue problem: A numerical study using the matrix-geometric solution

We consider a service system with two similar servers in which a customer, on arrival, joins the shorter queue. The state of the system is described by a pair (i,j), j?i, where i and j are the number of customers in the shorter and the longer queue, respectively. The stationary probability vector and several performance characteristics are obtained using the matrix-geometric solution technique.     

Dynamic scheduling of a GI/GI/1+GI queue with multiple customer classes

We consider a dynamic control problem for a GI/GI/1+GI queue with multiclass customers. The customer classes are distinguished by their interarrival time, service time, and abandonment time distributions. There is a cost c _k >0 for every class k∈{1,2,…,N} customer that abandons the queue before receiving service. The objective is to minimize average cost by dynamically choosing which customer class the server should next serve each time the server becomes available (and there are waiting customers from at least two classes). It is not possible to solve this control problem exactly, and so we formulate an approximating Brownian control problem. The Brownian control problem incorporates the entire abandonment distribution of each customer class. We solve the Brownian control problem under the assumption that the abandonment distribution for each customer class has an increasing failure rate. We then interpret the solution to the Brownian control problem as a control for the original dynamic scheduling problem. Finally, we perform a simulation study to demonstrate the effectiveness of our proposed control.     

A Load-Balanced Network with Two Servers

A load-balanced network with two queues Q ₁ and Q ₂ is considered. Each queue receives a Poisson stream of customers at rate _i , i=1,2. In addition, a Poisson stream of rate arrives to the system; the customers from this stream join the shorter of two queues. After being served in the ith queue, i=1,2, customers leave the system with probability 1–p _i ^*, join the jth queue with probability p(i,j), j=1,2, and choose the shortest of two queues with probability p(i,{1,2}). We establish necessary and sufficient conditions for stability of the system.     

Stochastic matrices and a property of the infinite sequences of linear functionals

Our starting point is the proof of the following property of a particular class of matrices. Let T={T_i,j} be a n×m non-negative matrix such that ∑_jT_i,j=1 for each i. Suppose that for every pair of indices (i,j), there exists an index l such that T_i,l≠T_j,l. Then, there exists a real vector k=(k₁,k₂,…,k_m)^T,k_i≠k_j,i≠j;0<k_i?1, such that, if i≠j.Then, we apply that property of matrices to probability theory. Let us consider an infinite sequence of linear functionals , corresponding to an infinite sequence of probability measures {μ_(·)(i)}_i∈N, on the Borel σ-algebra such that, . The property of matrices described above allows us to construct a real bounded one-to-one piecewise continuous and continuous from the left function f such that

     

Decomposable bilinear numerical radii

Let A be an n-square normal matrix over $\text{C}$, and Q_{m, n} be the set of strictly increasing integer sequences of length m chosen from 1,…, n. For α,β∈Q_{m, n} denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k∈{0,1,…,m} write z.sfnc;α∩β|=k if there exists a rearrangement of 1,…,m, say i₁,…,i_k, i_k+1,…,i_m, such that α(i_j)=β(i_j), j=1,…,k, and {α(i_k+1),…,α(i_m)};∩{β(i_k+1),…,β(i_m)}=ø. Let

be the group of n-square unitary matrices. Define the nonnegative number

$\text{?}{}_{\mathrm{k}}\text{(A)=}\text{max}\text{U∈}\textcolor{red}{}\text{|}\text{det}\text{(U}{}^1\text{AU) [α|β]|}$

, where |α∩β|=k. Theorem 1 establishes a bound for ?_k(A), 0?k<m?1, in terms of a classical variational inequality due to Fermat. Let A be positive semidefinite Hermitian, n?2m. Theorem 2 leads to an interlacing inequality which, in the case n=4, m=2, resolves in the affirmative the conjecture that

$\text{?}{}_{\mathrm{m}}\text{(A)??}{}_{\mathrm{m?1}}\text{(A)????}{}_0\text{(A)}$

.     

A queueing network with a single cyclically roving server

A queueingnetwork that is served by asingle server in a cyclic order is analyzed in this paper. Customers arrive at the queues from outside the network according to independent Poisson processes. Upon completion of his service, a customer mayleave the network, berouted to another queue in the network orrejoin the same queue for another portion of service. The single server moves through the different queues of the network in a cyclic manner. Whenever the server arrives at a queue (polls the queue), he serves the waiting customers in that queue according to some service discipline. Both the gated and the exhaustive disciplines are considered. When moving from one queue to the next queue, the server incurs a switch-over period. This queueing network model has many applications in communication, computer, robotics and manufacturing systems. Examples include token rings, single-processor multi-task systems and others. For this model, we derive the generating function and the expected number of customers present in the network queues at arbitrary epochs, and compute the expected values of the delays observed by the customers. In addition, we derive the expected delay of customers that follow a specific route in the network, and we introduce pseudo-conservation laws for this network of queues.Summary of notation B_i, B _i ^* (s) service time of a customer at queue i and its LST - b_i, b_i ⁽²⁾ mean and second moment of B_i - R_i, R _i ^* (s) duration of switch-over period from queue i and its LST - r_i, r_i mean and second moment of R_i - r, r⁽²⁾ mean and second moment of _i ^N =₁R_i - _i external arrival rate of type-i customers - _i total arrival rate into queue i - _i utilization of queue i; _i=_i - system utilization _i ^N =_1i - c=E[C] the expected cycle length - X _i ^j number of customers in queue j when queue i is polled - X_i=X _i ⁱ number of customers residing in queue i when it is polled - f_i(j) - X _i ^* number of customers residing in queue i at an arbitrary moment - Y_i the duration of a service period of queue i - W_i,T_i the waiting time and sojourn time of an arbitary customer at queue i - F^*(z₁, z₂,..., z_N) GF of number of customers present at the queues at arbitrary moments - F_i(z₁, z₂,..., z_N) GF of number of customers present at the queues at polling instants of queue i - ¯F_i(z₁, z₂,...,z_N) GF of number of customers present at the queues at switching instants of queue i - V_i(z₁, z₂,..., z_N) GF of number of customers present at the queues at service initiation instants at queue i - ¯V_i(z₁,z₂,...,z_N) GF of number of customers present at the queues at service completion instants at queue i The work of this author was supported by the Bernstein Fund for the Promotion of Research and by the Fund for the Promotion of Research at the Technion.Part of this work was done while H. Levy was with AT&T Bell Laboratories.     

Competitive online scheduling of perfectly malleable jobs with setup times

We study how to efficiently schedule online perfectly malleable parallel jobs with arbitrary arrival times on m ? 2 processors. We take into account both the linear speedup of such jobs and their setup time, i.e., the time to create, dispatch, and destroy multiple processes. Specifically, we define the execution time of a job with length p_j running on k_j processors to be p_j/k_j + (k_j − 1)c, where c > 0 is a constant setup time associated with each processor that is used to parallelize the computation. This formulation accurately models data parallelism in scientific computations and realistically asserts a relationship between job length and the maximum useful degree of parallelism. When the goal is to minimize makespan, we show that the online algorithm that simply assigns k_j so that the execution time of each job is minimized and starts jobs as early as possible has competitive ratio 4(m − 1)/m for even m ? 2 and 4m/(m + 1) for odd m ? 3. This algorithm is much simpler than previous offline algorithms for scheduling malleable jobs that require more than a constant number of passes through the job list.     

Optimal utilization of service facility for ak-out-of-n system with repair by extending service to external customers in a retrial queue

In this paper, we study ak-out-of-n system with single server who provides service to external customers also. The system consists of two parts: (i) a main queue consisting of customers (failed components of thek-out-of-n system) and (ii) a pool (of finite capacityM) of external customers together with an orbit for external customers who find the pool full. An external customer who finds the pool full on arrival, joins the orbit with probability γ and with probability 1- γ leaves the system forever. An orbital customer, who finds the pool full, at an epoch of repeated attempt, returns to orbit with probability δ (< 1) and with probability 1- δ leaves the system forever. We compute the steady state system size probability. Several performance measures are computed, numerical illustrations are provided.     

Self-adjusting stochastic processes

Let A_t(i, j) be the transition matrix at time t of a process with n states. Such a process may be called self-adjusting if the occurrence of the transition from state h to state k at time t results in a change in the hth row such that A_t+1(h, k) ? A_t(h, k). If the self-adjustment (due to transition h → kx) is A_{t + 1}(h, j) = λA_t(h, j) + (1 ? λ)δ_jk (0 < λ < 1), then with probability 1 the process is eventually periodic. If A₀(i, j) < 1 for all i, j and if the self-adjustment satisfies A_{t + 1}(h, k) = ?(A_t(h, k)) with ?(x) twice differentiable and increasing, x < ?(x) < 1 for 0 ? x < 1,?(1) = ?′(1) = 1, then, with probability 1, lim A_t does not exist.     

Infinite Server Queues with Synchronized Departures Driven by a Single Point Process

We analyze an infinite-server queueing model with synchronized arrivals and departures driven by the point process {T _n } according to the following rules. At time T _n , a single customer (or a batch of size β _n ) arrives to the system. The service requirement of the ith customer in the nth batch is σ _i,n . All customers enter service immediately upon arrival but each customer leaves the system at the first epoch of the point process {T _n } which occurs after his service requirement has been satisfied. For this system the queue length process and the statistics of the departing batches of customers are investigated under various assumptions for the statistics of the point process {T _n }, the incoming batch sequence {β _n }, and the service sequence {σ _i,n }. Results for the asymptotic distribution of the departing batches when the service times are long compared to the interarrival times are also derived.     

Nonsingularity of matrices associated with classes of arithmetical functions on lcm-closed sets

Let S = {x₁, … , x_n} be a set of n distinct positive integers and f be an arithmetical function. Let [f(x_i, x_j)] denote the n × n matrix having f evaluated at the greatest common divisor (x_i, x_j) of x_i and x_j as its i, j-entry and (f[x_i, x_j]) denote the n × n matrix having f evaluated at the least common multiple [x_i, x_j] of x_i and x_j as its i, j-entry. The set S is said to be lcm-closed if [x_i, x_j] ∈ S for all 1 ? i, j ? n. For an integer x > 1, let ω(x) denote the number of distinct prime factors of x. Define ω(1) = 0. In this paper, we show that if S = {x₁, … , x_n} is an lcm-closed set satisfying , and if f is a strictly increasing (resp. decreasing) completely multiplicative function, or if f is a strictly decreasing (resp. increasing) completely multiplicative function satisfying (resp. f(p) ? p) for any prime p, then the matrix [f(x_i, x_j)] (resp. (f[x_i, x_j])) defined on S is nonsingular. By using the concept of least-type multiple introduced in [S. Hong, J. Algebra 281 (2004) 1-14], we also obtain reduced formulas for det(f(x_i, x_j)) and det(f[x_i, x_j]) when f is completely multiplicative and S is lcm-closed. We also establish several results about the nonsingularity of LCM matrices and reciprocal GCD matrices.     

Large Closed Queueing Networks in Semi-Markov Environment and Their Application

The paper studies closed queueing networks containing a server station and k client stations. The server station is an infinite server queueing system, and client stations are single-server queueing systems with autonomous service, i.e. every client station serves customers (units) only at random instants generated by a strictly stationary and ergodic sequence of random variables. The total number of units in the network is N. The expected times between departures in client stations are (N μ _j )⁻¹. After a service completion in the server station, a unit is transmitted to the jth client station with probability p _j (j=1,2,…,k), and being processed in the jth client station, the unit returns to the server station. The network is assumed to be in a semi-Markov environment. A semi-Markov environment is defined by a finite or countable infinite Markov chain and by sequences of independent and identically distributed random variables. Then the routing probabilities p _j (j=1,2,…,k) and transmission rates (which are expressed via parameters of the network) depend on a Markov state of the environment. The paper studies the queue-length processes in client stations of this network and is aimed to the analysis of performance measures associated with this network. The questions risen in this paper have immediate relation to quality control of complex telecommunication networks, and the obtained results are expected to lead to the solutions to many practical problems of this area of research.      

A critically loaded multiclass Erlang loss system

We consider a generalization of the classical Erlang loss model to multiple classes of customers. Each of the J customer classes has an associated Poisson arrival process and an arbitrary holding time distribution. Classj customers requireM _j servers simultaneously. We determine the asymptotic form of the blocking probabilities for all customer classes in the regime known as critical loading, where both the number of servers and offered load are large and almost equal. Asymptotically, the blocking probability of classj customers is proportional toM _j . This asymptotic result provides an approximation for the blocking probabilities which is reasonably accurate. We also consider the behavior of the Erlang fixed point (reduced load approximation) for this model under critical loading. Although the blocking probability of classj customers given by the Erlang fixed point is again asymptotically proportional toM _j , the Erlang fixed point typically gives the wrong limit. Next we show that under critical loading the throughputs have a pleasingly simple form of monotonicity with respect to arrival rates: the throughput of classi is increasing in the arrival rate of classi and decreasing in the arrival rate of classj forji. Finally, we compare two simple control policies for this system under critical loading.     

A variable projection method for solving separable nonlinear least squares problems

Consider the separable nonlinear least squares problem of findinga εR ⁿ and α εR ^k which, for given data (y _i ,t _i ),i=1,2,...m, and functions ? _j (α,t),j=1,2,...,n(m>n), minimize the functional $$r(a,\alpha ) = \left\| {y - \Phi (\alpha )a} \right\|_2^2$$ where θ(α) _ij =? _j (α,t _i ). Golub and Pereyra have shown that this problem can be reduced to a nonlinear least squares problem involvingα only, and a linear least squares problem involvinga only. In this paper we propose a new method for determining the optimalα which computationally has proved more efficient than the Golub-Pereyra scheme.     

Evaluating the impact of operating conditions on the performance of appointment scheduling rules in service systems

The general practice in implementing an appointment scheduling rule (ASR) is to enforce a certain rule, such as “block appointment”, to schedule customer arrivals in service systems. The operating environments of service systems are expected to affect considerably the performance of a selected ASR. The objective of this paper is to evaluate the impact of the environmental factors which include probability of no-show (ρ), the coefficient of variation (C_ν) of service times, and the number of customers per service session (N). The extent to which a certain environmental factor affects the performance of ASR is examined to see if there is any ASR that performs well under most operating conditions. Under situations characterized by 27 different combinations of the factors ρ, C_ν, and N, the performance of nine scheduling rules are evaluated by a simulation study. The simulation results show that an ASR designed to reduce customer waiting time performs very well in most operating environments considered. One commonly used ASR in real-world service systems, which schedules several customers to arrive at the start of each service session, tends to induce long customer waiting time.     

A subdeterminant inequality for normal matrices

Let A be an n × n normal matrix over $\text{C}$, and Q_m, _n be the set of strictly increasing integer sequences of length m chosen from 1,…,n. For α, β ? Q_m, _n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k ? {0, 1,…, m} we write |α∩β| = k if there exists a rearrangement of 1,…, m, say i₁,…, i_k, i_k+1,…, i_m, such that α(i_j) = β(i_j), i = 1,…, k, and {α(i_k+1),…, α(i_m) } ∩ {β(i_k+1),…, β(i_m) } = ?. A new bound for |detA[α|β ]| is obtained in terms of the eigenvalues of A when 2m = n and |α∩β| = 0.Let $\text{U}$_n be the group of n × n unitary matrices. Define the nonnegative number

where | α ∩ β| = k. It is proved that

Let A be semidefinite hermitian. We conjecture that ρ₀(A) ? ρ₁(A) ? ··· ? ρ_m(A). These inequalities have been tested by machine calculations.     

Approximating reversal distance for strings with bounded number of duplicates

For a string A=a₁…a_n, a reversalρ(i,j), 1?i?j?n, transforms the string A into a string A^′=a₁…a_i-1a_ja_j-1…a_ia_j+1…a_n, that is, the reversal ρ(i,j) reverses the order of symbols in the substring a_i…a_j of A. In the case of signed strings, where each symbol is given a sign + or -, the reversal operation also flips the sign of each symbol in the reversed substring. Given two strings, A and B, signed or unsigned, sorting by reversals (SBR) is the problem of finding the minimum number of reversals that transform the string A into the string B.Traditionally, the problem was studied for permutations, that is, for strings in which every symbol appears exactly once. We consider a generalization of the problem, k-SBR, and allow each symbol to appear at most k times in each string, for some k?1. The main result of the paper is an O(k²)-approximation algorithm running in time O(n). For instances with , this is the best known approximation algorithm for k-SBR and, moreover, it is faster than the previous best approximation algorithm.     

Economic application in a finite capacity multi-channel queue with second optional channel

We consider a finite capacity M/M/R queue with second optional channel. The interarrival times of arriving customers follow an exponential distribution. The service times of the first essential channel and the second optional channel are assumed to follow an exponential distribution. As soon as the first essential service of a customer is completed, a customer may leave the system with probability (1 − θ) or may opt for the second optional service with probability θ (0 ? θ ? 1). Using the matrix-geometric method, we obtain the steady-state probability distributions and various system performance measures. A cost model is established to determine the optimal solutions at the minimum cost. Finally, numerical results are provided to illustrate how the direct search method and the tabu search can be applied to obtain the optimal solutions. Sensitivity analysis is also investigated.