# Modelling the performance of computer mirroring with difference queues

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2 is recognized as a Poisson process. Tasks scheduled for execution are placed (arrive) in the queue. Tasks are executed (served) in order of arrivals. This queueing model is the well known model of the single server Poisson queue M/M/. It is defined Main Arrivals q S Departures S q Backup Difference queue Figure. Queueing model of computer mirroring system. with two parameters: the average arrival rate λ and the average service (processing) rate. The probability density functions for the interarrival and service times are exponential. The utilization of the system is defined as ρ = λ. Of particular interest for the discussion that follows is the probability distribution of the waiting time (including service) for the M/M/ queue: ( λ) t f ( t) = ( λ ) e. () The mirroring system consists of two computer system units, each comprised of a queue and an execution unit. The tasks are scheduled (arrive) for both systems simultaneously and the arrival rate is modelled after the Poisson process. Execution of tasks on each of the computers is independent from the other. The average arrival rate is λ = λ = λ and the service rates are and respectively. This system is very similar to the parallel network of queues described by Saaty [] except that the same Poisson process generates arrivals to both queues. When referring to the lengths L, L of queues q and q we shall account for the tasks being served, i.e. we will use the total number of items in the queueing system. The statistical equilibrium equations for the steady state behaviour of the model of the mirroring system are difficult to solve in closed form. However the solution can be derived easily using the approach of difference queues. 3. DIFFERENCE QUEUES The concept of a difference queue naturally follows from the queueing model of the mirroring system. During the parallel operation of two computer systems it is expected that the execution times for the individual tasks, and therefore queue lengths on each of the systems will differ slightly. Since in our analysis of the performance of the mirroring system we are interested in the delay of completion of a task on the backup system, we would like to estimate how much time is needed for the backup computer to catch up with the main system in case it has fallen behind. The backup system needs to execute the tasks that remain in its queue, and which have already departed (have been processed) from the queue in the main computer. Thus the concept of the difference between two queues arises. It is interesting to note that the difference between the two queues defined above is also a queue. This queue, because of its origin, will be called the difference queue, or the d-queue. The arrivals to the

4 For modelling such a scenario we will use a d- queue type 3. The arrival for the d-queue is still the same as the departure from the queue q on the main system, however the arrival rate is no longer defined as the processing rate of the main system. A new arrival to the d-queue occurs only when the queue on the main system is not empty. Recall that the probability of the M/M/ queue to be not empty is P(NOT empty)= ρ. Consequently the probability of the new arrival to the d-queue over the interval t, t + δ t, to the first order, is ρ δ t. The ( ) departures from the d-queue are the Poisson departures from the q on the backup system with average service rate of. The probability of a departure is δ t. The stationary distribution can be derived from the following equilibrium equations ρ P = 0 P ( ρ + ) P = ρ P + 0 P... ( ρ + ) P n = ρ P n + P n + (4) where P i is the probability of having i elements in the type 3 d-queue. Substituting ρ = we get a set of equilibrium equations defining the M/M/ Poisson queue [, page 40] with the arrival rate = ρ and service rate. Hence the probability density function for the delay in the mirroring system (i.e. waiting time in the type 3 d-queue) is ( ρ ) tl f( tl) = ( ρ ) e, (5) and the mean is E[ t L ] =, ρ <. (6) ρ Note: Equations 5, 6 become equations, 3 for ρ =. 5. DISCUSSION A typical mirroring system with identical main and backup computers may experience significant delays in the operation of the backup system under moderate to high load conditions. When = and ρ is close to the average delay defined in Equation 6 is high. For ρ = we have no steady state solution, and consequently no limits on the delays for the system defined in our model. The performance of the mirroring system can be improved by increasing the processing rate of the backup system and decreasing the utilization ρ of the main system so that ρ <. (7) In fact since ρ = λ and ρ = λ in Equations 5 and 6 the distribution of delays in the mirroring system under low load conditions ( ρ < ) depends only on the arrival rate λ = λ = λ and the processing rate of the backup system. (Note that the backup system need not to be faster than the main system.) Changing the processing rate of the main system has no effect on the delay jitter in the mirroring system (steady state) under low load. The relative speeds of the main and backup systems influence the delay in computer mirroring only under the heavy load operation (ρ ). When the main system is continuously busy the delays are inversely proportional to the difference in processing rates. The k th percentile for the delay time T 0 derived from the density function in Equation 5 is 00 ln T 00 k 0 =. (8) ρ For example: a computer mirroring system consisting of two computers with identical processing rates = = 00 transactions per second, and the utilization of the main computer of 70%, ρ =0.7, would experience delay of less than 0.54 seconds 99% of the time. If the utilization of the main system was increased to ρ =0.9, the delay of less than 4.6 seconds should be expected 99% of the time. Note that % of the time the delay may exceed the average processing time 460 times. 6. CONCLUSION Performance of the computer mirroring system is directly linked to the behaviour of the d-queue, the difference queue derived from the queues on the main and backup computers. The steady state solution for the model exists for ρ < or λ <. Minimum delays in the operation of the backup system are obtained for λ << under normal operating conditions (low load), and << for operation under high load. 7. REFERENCES

5 . Cox, D. R., Smith, W. L., Queues, Methuen & Co Ltd, Saaty, T. L., Elements of Queueing Theory, McGraw-Hill, White, J. A., Schmidt, J.W., Bennett, G.K., Analysis of Queueing Systems, Academic Press, 975.

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Simulation Software 1 Introduction The features that should be programmed in simulation are: Generating random numbers from the uniform distribution Generating random variates from any distribution Advancing