Closed Queuing Networks (Mean Value Analysis)

1. Closed Queuing Networks (Mean Value Analysis)

2. Closed Queuing Networks • Arise in two situations • When “source” of requests is explicitly modeled. That means, there is one node that models the clients. • When there is a limit on the number of requests that can circulate in a server network, and the assumption is that this limit has been reached and being maintained due to a high load. • (Implicitly, there is a queue at the “entrance” to the server network which is always non-empty).

3. Consider the “client-server” situation (class example)

4. Parameters of the queuing network • For this queuing network, the parameters are: • Load parameters: Number of clients (M), client think time (EXP(lambda)) • Think time is assumed to be exponential • Server parameters: Routing matrix (X), service rates (mu_i) for all the servers • Service time is assumed to be exponential

5. Metrics of the queuing network • Given server parameters, and think timr we may be interested in: • How many clients can the server system support? • What is the maximum achievable throughput of the system? • What is the bottleneck server?

6. Analysis Technique • Calculate relative visit counts • Apply Mean Value Analysis

7. Relative Visit Counts • Since there is no actual notion of “exit” from the network, in the case of closed QNs, we cannot really find “expected number of visits until exit” from the network • We can only talk about the number of visits made to a server Si relative to the number of visits made to a server Sj • Thus visit counts in closed QNs are called relative visit counts

8. Relative Visit Counts • Suppose V_i ‘s are the expected number of visits made to node i during a certain observation period. • Then as explained in class, the equation • V = V X • expresses the relationships between the visit counts • This is a linearly dependent set of equations

9. Relative Visit Counts • Thus, we can choose to express V_i’s in terms of one “chosen” V_j. • In the example given, it makes sense for this chosen V_j to be V_C, the visit count to the client node • Further, if we set the value for this V_C to 1, the relative visit counts to the other servers have a good “explanation” • They are the number of visits made to the server nodes by a request from a client, before going back to the client node

10. Relative Visit Count Calculation • As done in class • Once we have the relative visit counts for the case of V_C=1, we can find response time, if we knew the per-visit response time of a request to a node

11. Arrival Theorem • For finding this, we use a very big result in queueing systems, called the “Arrival Theorem”. It states for a closed QN with “load level” M, a request arriving at a node i “sees” the state of the queue as if the queuing network has one less customer in the system. • In other words, it sees the state of the system as if it itself is not there in the system, but just observing the system

12. Mean Value Analysis: Terms • Each of the following are defined for a load level M • Ni(M) = average queue length at node i • Ti (M) = throughput at node i • Ri(M) = response time at node I • Tsys(M) = system throughput • Rsys(M) = Actual response time • Rcycle(M) = Cycle Time • Note that Tsys(M) and Rsys(M) are values that are relative to the visit counts. Other values are not relative to visit counts.

13. Mean Value Analysis • Assume that we know all the above quantities at level M-1. Can we derive them at M? Then we have a recursion. • Using arrival theorem, we can write the response time of a request arriving at node i as: • Ri(M) = [ 1 + Ni(M-1)] * tau_i • This is because arriving request is expected to “see” Ni(M-1) requests in front of it which must be processed before it, and then its own service time must be counted for calculating response time. • Note that the client node is an exception. For the client node • RC(M) = think_time, for all load levels M

14. MVA…contd • Now we have response time at each node, and we have the visit counts. Thus • Rcycle(M) = VC RC(M) + V0 R0(M) + V1 R1(M)+V2 R2(M) • Rsys (M) = V0 R0 (M) + V1 R1 (M) +V2 R2 (M) • Now, as explained in class, we apply little’s law to the whole system and get system throughput • Tsys (M) = M / Rcycle(M)

15. MVA…contd • Now we have system throughput at load level Tsys (M). We can get node throughput by multiplying by visit counts: • Ti (M) = Vi Tsys (M) • Now thet we have node throughput, and recalling that we first found response time at node i, we can apply Little’s Law at node level: • Ni (M) = Ti (M) * Ri (M) • Recursion is complete!

16. MVA…contd • So if we have to find, say, system response time at load level 3, we start with • Ni (0) = 0 and proceed with finding metrics at load levels 1, 2, 3 iteratively.

17. Example in Spreadsheet • Example in spreadsheet represents a server system which processes two types of requests: “orders” and complaints (“trouble tickets”) • If modeled as closed QN, model a node of M clients who are interacting with this system • Client issues either request for “order” or “trouble ticket” (w. p. pO, pT respectively) • These go to different web servers (W1, W2) • Request may have to visit application servers (A1, A2). Multiple such loops are possible. • Picture on next page

18. Example in Spreadsheet

19. Example in spreadsheet: activity for you • Calculate visit count based on the topology shown • Express each visit count in terms of Vc and set Vc to 1 • In the spreadsheet, all cells coloured yellow are input parameters. Everything else is calculated. • Check that the visit counts as coded in the spreadsheet match your calculations. • The MVA iteration is coded in the rows. Row i values are calculated using the Ni of Row i-1.