Performance analysis approximation in a queueing system of type M/G/1

In this work, we apply the strong stability method to obtain an estimate for the proximity of the performance measures in the M/G/1 queueing system to the same performance measures in the M/M/1 system under the assumption that the distributions of the service time are close and the arrival flows coincide. In addition to the proof of the stability fact for the perturbed M/M/1 queueing system, we obtain the inequalities of the stability. These results give with precision the error, on the queue size stationary distribution, due to the approximation. For this, we elaborate from the obtained theoretical results, the STR-STAB algorithm which we execute for a determined queueing system: M/Coxian − 2/1. The accuracy of the approach is evaluated by comparison with simulation results.     

Time-dependent analysis for a queue modeled by an infinite system of partial differential equations

By studying the spectrum of the underlying operator corresponding to the exhaustive-service M/G/1 queueing model with single vacations we prove that the time-dependent solution of the model strongly converges to its steady-state solution.     

The performance evaluation of a multi-stage JIT production system with stochastic demand and production capacities

This paper discusses a single-item, multi-stage, serial Just-in-Time (JIT) production system with stochastic demand and production capacities. The JIT production system is modeled as a discrete-time, M/G/1-type Markov chain. A necessary and sufficient condition, or a stability condition, under which the system has a steady-state distribution is derived. A performance evaluation algorithm is then developed using the matrix analytic methods. In numerical examples, the optimal numbers of kanbans are determined by the proposed algorithm. The optimal numbers of kanbans are robust for the variations in production capacity distribution and demand distribution.     

Asymptotic analysis of the M/G/1 queue with a time‐dependent arrival rate

We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p _n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some structural properties of a Markovian storage/production system

We consider a storage/production system with state-dependent production rate and state-dependent demand arrival rate. Every arriving demand gives rise to a 'peak' in the trajectory of the content process. We characterize the processes N ₀(x)and N₁(x), defined as the number of peaks and the number of record peaks, respectively, before the content reaches the level x. The results are applied to the virtual waiting time process W(t) of a M/G/1 queue. Assuming that W(0)= x₀, M(x) is defined to be the number of arrivals before the virtual waiting time drops from x₀ to x₀-x(0⩽ x ⩽ x₀) ; in particular, Mx₀)is the number of customers arriving during the first busy period. It is shown that (M(x))(0⩽ x ⩽ x₀)is a compound Poisson process, and its jump size distribution is derived in closed form. This revised version was published online in June 2006 with corrections to the Cover Date.