PERFORMANCE MODELLING OF COMPUTER SYSTEMS AND COMPUTER NETWORKS

1. PERFORMANCE MODELLING OF COMPUTER SYSTEMS AND COMPUTER NETWORKS Ramon Puigjaner Universitat de les Illes Balears Palma, Spain putxi@uib.es LANC 2007. San José, Costa Rica. October 2007. .

2. OUTLINE • INTRODUCTION • CONCEPT OF QUEUE • CONCEPT OF QUEUEING NETWORK • NUMERICAL TECHNIQUES • EXACT ANALYTICAL SOLUTIONS • APPROXIMATE ANALYTICAL SOLUTIONS • SIMULATION TECHNIQUES LANC 2007. San José, Costa Rica. October 2007.

3. INTRODUCTION • What is the performance of a Computer Network? • Performance is how a software is using a hardware when they are serving a load. LANC 2007. San José, Costa Rica. October 2007.

4. INTRODUCTION • This definition considers the three elements intervening in a system: • The load that is externally defined. • The hardware to be used. • The basic software that controls the hardware. LANC 2007. San José, Costa Rica. October 2007.

5. INTRODUCTION: Performance measures • The performance of a Computer Network is not a unique value but a set of them to take into account the heterogeneous composition of such kind of systems. • External performance measures • response time • throughput (flow through the system) • loss rate LANC 2007. San José, Costa Rica. October 2007.

6. INTRODUCTION: Performance measures • Internal performance measures • mean queue length • device utilisation (% of busy time) • overlap • overhead (operating system utilisation, paging, etc.) LANC 2007. San José, Costa Rica. October 2007.

7. INTRODUCTION: Performance tools • Measuring • Monitors • Logs • Hardware probes • Software probes • Modelling • Benchmarking LANC 2007. San José, Costa Rica. October 2007.

8. INTRODUCTION: Performance tools • Measuring • Modelling • Queuing networks • Petri nets • Markov chains • Benchmarking • Workload modelling LANC 2007. San José, Costa Rica. October 2007.

9. INTRODUCTION: Measuring • Measuring is the technique to be used when system is installed and running. It is used to verify whether the performance requirements are met or not. LANC 2007. San José, Costa Rica. October 2007.

10. INTRODUCTION: Modelling • A model is an abstract mathematical representation of the system behaviour in steady state. It is the appropriate technique when the computer network, partially or totally, does not exist. Main existing techniques are: • Petri nets • Better suited to represent synchronisation mechanisms • Solving techniques may be either numerical (based on Markov chains) or simulation. LANC 2007. San José, Costa Rica. October 2007.

11. INTRODUCTION: Modelling • Queuing networks • Better suited to represent customer-server mechanisms • Solving techniques may be either analytical (closed form formulae) or numerical (based on Markov chains) or simulation. • Markov chains • High abstraction level • Solving techniques are most frequently numerical. LANC 2007. San José, Costa Rica. October 2007.

12. OUTLINE • INTRODUCTION • CONCEPT OF QUEUE • CONCEPT OF QUEUEING NETWORK • NUMERICAL TECHNIQUES • EXACT ANALYTICAL SOLUTIONS • APPROXIMATE ANALYTICAL SOLUTIONS • SIMULATION TECHNIQUES LANC 2007. San José, Costa Rica. October 2007.

13. CONCEPT OF QUEUE • Queue: A customer that arrives and finds the server busy joins the queue • Service mechanism: It consists of one or more servers that give service to the customers from the queue LANC 2007. San José, Costa Rica. October 2007.

14. CONCEPT OF QUEUE • Customer source characteristics • finite or infinite • distribution of inter-arrival times between consecutive customer arrivals • customer service request • Service station characteristics • queue number and capacity • server number • server capacity • service discipline • queue policy LANC 2007. San José, Costa Rica. October 2007.

15. CONCEPT OF QUEUE • Single queue with single server • Single queue with single server with state dependent capacity LANC 2007. San José, Costa Rica. October 2007.

16. CONCEPT OF QUEUE • Single queue with multiple servers LANC 2007. San José, Costa Rica. October 2007.

17. CONCEPT OF QUEUE • Multi-server with no queue LANC 2007. San José, Costa Rica. October 2007.

18. CONCEPT OF QUEUE • Infinite server LANC 2007. San José, Costa Rica. October 2007.

19. OUTLINE • INTRODUCTION • CONCEPT OF QUEUE • CONCEPT OF QUEUEING NETWORK • NUMERICAL TECHNIQUES • EXACT ANALYTICAL SOLUTIONS • APPROXIMATE ANALYTICAL SOLUTIONS • SIMULATION TECHNIQUES LANC 2007. San José, Costa Rica. October 2007.

20. CONCEPT OF QUEUEING NETWORK • A queuing network is nothing else but a collection of single queues, which are arbitrarily interconnected. • A queuing network is an oriented graph that has in each node a server of some type. • The time in traversing the network is spent in the nodes and the arcs are traversed in a null time. LANC 2007. San José, Costa Rica. October 2007.

21. CONCEPT OF QUEUEING NETWORK • Queuing networks may be either open or closed. • In an open queuing network, customers arrive from outside, circulate through the nodes, and finally they depart from the network. • In a closed queuing network, there is a fixed number of customers constantly circulating through the nodes. Neither departures from the network nor arrivals to the network are allowed. • It is possible to have a queuing network which is both open and closed. Such a network is known as a mixed network. LANC 2007. San José, Costa Rica. October 2007.

22. EXAMPLES OF OPEN QUEUING NETWORKS • Tandem configuration LANC 2007. San José, Costa Rica. October 2007.

23. EXAMPLES OF OPEN QUEUING NETWORKS • Tree-like configuration LANC 2007. San José, Costa Rica. October 2007.

24. EXAMPLES OF OPEN QUEUING NETWORKS • Tree-like configuration LANC 2007. San José, Costa Rica. October 2007.

25. EXAMPLES OF CLOSED QUEUING NETWORKS • Cyclic network (closed tandem configuration) LANC 2007. San José, Costa Rica. October 2007.

26. EXAMPLES OF CLOSED QUEUING NETWORKS • Arbitrary configuration LANC 2007. San José, Costa Rica. October 2007.

27. EXAMPLES OF CLOSED QUEUING NETWORKS • Central server model LANC 2007. San José, Costa Rica. October 2007.

28. EXAMPLES OF CLOSED QUEUING NETWORKS • Central server model LANC 2007. San José, Costa Rica. October 2007.

29. EXAMPLES OF CLOSED QUEUING NETWORKS • Central server model LANC 2007. San José, Costa Rica. October 2007.

30. EXAMPLES OF MIXED QUEUING NETWORKS LANC 2007. San José, Costa Rica. October 2007.

31. CONCEPT OF QUEUEING NETWORK • Observations • Each node can have any of the single-node characteristics described above. • In order to specify the queuing network we need to provide information concerning the routing; that is to specify how a customer chooses the next node when it leaves the current node. This routing can be deterministic, probabilistic, function of the state, etc. LANC 2007. San José, Costa Rica. October 2007.

32. CONCEPT OF QUEUEING NETWORK • How to set-up a queuing network model? • The notion of customer • Typically a customer may be a piece of software in a computer system, an information packet in a packet-switched environment, a phone call in a circuit-switched environment, etc. • Customer classes will be defined if there are differences in the resource consumption or in the routing across the network LANC 2007. San José, Costa Rica. October 2007.

33. CONCEPT OF QUEUEING NETWORK • How to set-up a queuing network model? • The notion of node • A node is a service mechanism that may be a hardware component or a piece of software or a combination of both, e.g. a CPU, a disk, a memory module, a bus, a trunk, a switching node, etc. • Each service mechanism has a buffer (the queue), where customers wait until they are served. The buffer capacity is finite; that is, they can accommodate a finite number of customers. However, if a finite buffer has low probability of being full, then it can be assumed as infinite. LANC 2007. San José, Costa Rica. October 2007.

34. CONCEPT OF QUEUEING NETWORK • How to set-up a queuing network model? • Collecting information • Once customers and server have been identified, it is necessary to characterise service time distributions at each node, routing probabilities and inter-arrival time distributions. • In many cases, this information can be compiled from raw data (technical information, measurements, etc.); in other cases it is based on an educated guess. LANC 2007. San José, Costa Rica. October 2007.

35. CONCEPT OF QUEUEING NETWORK • Solution techniques for queuing networks To study the steady state behavior of a network the following techniques can be used: • Analytic solutions • Numerical techniques • Simulation techniques LANC 2007. San José, Costa Rica. October 2007.

36. OUTLINE • INTRODUCTION • CONCEPT OF QUEUE • CONCEPT OF QUEUEING NETWORK • NUMERICAL TECHNIQUES • EXACT ANALYTICAL SOLUTIONS • APPROXIMATE ANALYTICAL SOLUTIONS • SIMULATION TECHNIQUES LANC 2007. San José, Costa Rica. October 2007.

37. NUMERICAL TECHNIQUES • The behaviour of a queuing network can be described in terms of linear equations (known as the steady-state Kolmogorov equations). These equations can be solved numerically to obtain the solution. LANC 2007. San José, Costa Rica. October 2007.

38. NUMERICAL TECHNIQUES • To highlight this approach, let us consider the following two-node closed queuing network • Let us assume that: • there are 5 customers in the system. • µ1 and µ2 are the service rates. • both services are exponentially distributed. LANC 2007. San José, Costa Rica. October 2007.

39. NUMERICAL TECHNIQUES • The state of the system is described by (n1, n2), that there are the number of customers in each queue. • The numerical analysis approach involves the following steps: • Generation of all feasible states. • Setting-up the rate matrix. • Solving the steady state equations. LANC 2007. San José, Costa Rica. October 2007.

40. NUMERICAL TECHNIQUES • Generation of all feasible states. The states for our example are: (5,0) (4,1) (3,2) (2,3) (1,4) (0,5) LANC 2007. San José, Costa Rica. October 2007.

41. NUMERICAL TECHNIQUES • Setting-up the rate matrix. • This matrix which contains all the transitions and their associated rates between each pair of states. (5,0) (4,1) (3,2) (2,3) (1,4) (0,5) (5,0) * µ1 (4,1) µ2 * µ1 (3,2) µ2 * µ1 (2,3) µ2 * µ1 (1,4) µ2 * µ1 (0,5) µ2 * LANC 2007. San José, Costa Rica. October 2007.

42. NUMERICAL TECHNIQUES • Setting-up the rate matrix. • Let us refer to the this matrix as Q. • All blanks are assumed to be zero. • Each diagonal element marked with * is equal to the negative sum of the off-diagonal elements of the same row. LANC 2007. San José, Costa Rica. October 2007.

43. NUMERICAL TECHNIQUES • Solving the steady state equations. • Let p(n1, n2)be the steady-state probability that the system is in state (n1, n2)and P the row vector of these probabilities. To determine them we must solve the following system of equations: P x Q = 0 together with the condition LANC 2007. San José, Costa Rica. October 2007.

44. NUMERICAL TECHNIQUES • Solving the steady state equations. • From the knowledge of these probabilities we can determine performance measures such as: • Server utilisation: r1 = p(5,0) + p(4,1) + p(3,2) + p(2,3) + p(1,4): r2 = p(4,1) + p(3,2) + p(2,3) + p(1,4) + p(0,5) • Throughputs: l1 = r1x µ1 l2 = r2 x µ2 • Queue lengths: N1 = 5p(5,0) + 4p(4,1) + 3p(3,2) + 2p(2,3) + p(1,4) N2 = p(4,1) + 2p(3,2) + 3p(2,3) + 4p(1,4) + 5p(0,5) LANC 2007. San José, Costa Rica. October 2007.

45. NUMERICAL TECHNIQUES • Solving the steady state equations. • Advantages/disadvantages • There are packages, like QNAP2, that automatically set-up the rate matrix Q, solve it to find the P vector and give the performance results. Other packages give the vector P if the user is able to create the matrix Q. • This numerical technique gives the exact solution. There are also approximated solutions in some cases in order to reduce the amount of computation. • The approach is limited to cases where the number of states is not very large. • In queuing networks, quite often, the rate matrix is sparse. In this cases, one can analyse larger systems by using compact storage techniques. LANC 2007. San José, Costa Rica. October 2007.

46. OUTLINE • INTRODUCTION • CONCEPT OF QUEUE • CONCEPT OF QUEUEING NETWORK • NUMERICAL TECHNIQUES • EXACT ANALYTICAL SOLUTIONS • APPROXIMATE ANALYTICAL SOLUTIONS • SIMULATION TECHNIQUES LANC 2007. San José, Costa Rica. October 2007.

47. EXACT ANALYTICAL SOLUTIONS • An analytical solution means that we can obtain the probabilities of the steady steady by the application of a closed formula. • This formula will obviously be a function of the parameters of the system. • Quite often an analytic solution is so complicated that we can not evaluate it "on the back of an envelope". In fact, one might need to write a fairly sophisticated program. LANC 2007. San José, Costa Rica. October 2007.

48. EXACT ANALYTICAL SOLUTIONS • A certain class of queuing networks has an analytic solution, known as a product-form solution because the steady state probability has the form of the product of the state probabilities of each node. • Its solution can be easily evaluated. LANC 2007. San José, Costa Rica. October 2007.

49. EXACT ANALYTICAL SOLUTIONS • Product-form networks have been proved to be very useful in computer and communication systems performance modelling. • Also, there are a lot of queuing networks which do not have product-form solutions. These networks are analysed approximately. LANC 2007. San José, Costa Rica. October 2007.

50. EXACT ANALYTICAL SOLUTIONS • The BCMP theorem • It is the general theorem concerning queuing networks with product-form solutions. • Let us consider a BCMP queuing network with: • N nodes arbitrarily linked. • Multiple classes of customers • Probabilistic routing • External arrivals with state-dependent rates • Different service mechanisms LANC 2007. San José, Costa Rica. October 2007.