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Unformatted text preview: Probability and Statistics with Reliability, Queuing and Computer Science Applications Second edition by K.S. Trivedi Publisher-John Wiley & Sons Chapter 8 (Part 7) :Continuous Time Markov Chain Stochastic Petri Nets (Analysis and Applications) Dept. of Electrical & Computer engineering Duke University Email:kst@ee.duke.edu URL: www.ee.duke.edu/~kst Copyright © 2003 by K.S. Trivedi Copyright 1 Outline • Traditional methodology: (continuous-time) • • • • • Markov chains Introduction to stochastic Petri nets (SPN) Example of Wireless cellular system Example of a Multiprocessor System SPNP: A Software Package Recent research on SPN Copyright © 2003 by K.S. Trivedi Copyright 2 Traditional Methodology Study the system Packet, Cell, etc. Network Element: Router, Switch, etc. Buffer Copyright © 2003 by K.S. Trivedi Copyright 3 Traditional Methodology Step 1: Abstract the system λ n µ M/D/1/n Copyright © 2003 by K.S. Trivedi Copyright 4 Traditional Methodology Step 2: Construct Markov chain (CTMC) (2-dimensional) Note: m-stage Erlang (Em) is used to approximate deterministic service time (D). Copyright © 2003 by K.S. Trivedi Copyright 5 Traditional Methodology Step 3-a: Build system of linear equations (Steady-state solution) πQ = 0 π1 = 1 (where 1 = [1,1, .., 1]T) π: steady-state probability (row) vector Q: infinitesimal generator matrix Copyright © 2003 by K.S. Trivedi Copyright 6 Traditional Methodology Step 3-b: Or, build a system of linear, firstorder, ordinary differential equations (transient solution) dπ(t)/dt = π(t) Q (given π(0) ) π(t): state probability (row) vector; note: we used P(t) earlier for this vector Q: infinitesimal generator matrix Copyright © 2003 by K.S. Trivedi Copyright 7 Traditional Methodology Step 4a: If possible, find a closed form solution to the equation Step 4b: else numerically solve the equations Step 5: Calculate measures of interest Steps 1-3 , 4a and 5 are commonly handled manually. This is a rather tedious and error-prone procedure, especially when the number of states becomes very large. Copyright © 2003 by K.S. Trivedi Copyright 8 Can we find a new methodology to let computer do the tedious work? Copyright © 2003 by K.S. Trivedi Copyright 9 Stochastic Petri Net (SPN) Introduced in 1980s by Natkin, Florin, Molloy, Ajmone Marsan, Balbo, Conte, Bobbio, Trivedi, & others A modeling formalism for the automated generation and solution of Markovian stochastic systems. Copyright © 2003 by K.S. Trivedi Copyright 10 Petri Nets Petri Net (PN) is a graphical paradigm for the formal description of the logical interactions among parts or of the flow of activities in complex systems. PN are particularly suited to model: Concurrency and Conflict; Sequencing, conditional branching and looping Synchronization; Sharing of limited resources and Mutual exclusion. Copyright © 2003 by K.S. Trivedi Copyright 11 Petri Nets Petri Nets (PN) originated from the PhD thesis of Carl Adam Petri in 1962. A web service on PN is managed by the University of Aarhus in Denmark, where a bibliography with more that 7,800 items can be found. http://www.daimi.au.dk/PetriNets/ There are two regular International Conferences on PN: ATPN - Application and Theory of PN PNPM – PN and Performance Models Copyright © 2003 by K.S. Trivedi Copyright 12 Petri Nets The original PN did not have the notion of time. For performance and availability analysis it is necessary to introduce duration of events associated with PN transitions. For this reason, PN were subsequently extended to timed events following two main lines: Random durations : Stochastic PN (SPN) Deterministic or interval: Copyright © 2003 by K.S. Trivedi Copyright Timed PN (TPN) 13 Introduction to Petri Nets & Stochastic Petri Nets Copyright © 2003 by K.S. Trivedi Copyright 14 Definitions • A Petri net (PN) is a bipartite directed graph bipartite consisting of two kinds of nodes: places and places transitions transitions – Places typically represent conditions within the system being modeled – Transitions represent events occurring in the system that may cause change in the condition of the system – Arcs connect places to transitions and transitions to places But never an arc from a place to a place or from a transition to a transition Copyright © 2003 by K.S. Trivedi Copyright 15 Definition of PN A PN is a 5-tuple (P,T,I,O,M) P T I O M set of places set of transitions input arcs output arcs initial marking Copyright © 2003 by K.S. Trivedi Copyright 16 Basic Components of PN input place transition output place token input arc output arc Copyright © 2003 by K.S. Trivedi Copyright 17 PN Definitions • Input arcs are directed arcs drawn from places to transitions, representing the conditions that need to be satisfied for the event to be activated • Output arcs are directed arcs drawn from transitions to places, representing the conditions resulting from the occurrence of the event Copyright © 2003 by K.S. Trivedi Copyright 18 PN Definitions • Input places of a transition are the set of places that are connected to the transition through input arcs • Output places of a transition are the set of places to which output arcs exist from the transition Copyright © 2003 by K.S. Trivedi Copyright 19 PN Definitions • Tokens are dots (or integers) associated with places; A place containing tokens indicates that the corresponding condition holds n • Marking of a Petri net is a vector listing the number of tokens in each place of the net (n1 n2 … nP), P = # of Places Copyright © 2003 by K.S. Trivedi Copyright 20 PN Definitions • When input places of a transition have the required number of tokens, the transition is enabled. • An enabled transition may fire (event happens) taking a specified number of tokens from each input place and depositing a specified number of tokens in each of its output place. Copyright © 2003 by K.S. Trivedi Copyright 21 Example of a PN t1 p2 p1 . p1 – resource idle p2 – resource busy t1 – task arrives t2 – task completes t2 Copyright © 2003 by K.S. Trivedi Copyright 22 Example of a PN p3 t1 p1 p2 t2 p1 – resource is idle p2 – resource is busy t1 – task arrives t2 – task completes p3 – users Copyright © 2003 by K.S. Trivedi Copyright 23 The firing rules of a PN m tk m' Copyright © 2003 by K.S. Trivedi Copyright 24 Enabling & Firing of Transitions up t_repair “t_fail” fires t_fail down up t_repair “t_repair” fires “t_fail” fires up t_fail t_repair down “t_repair” fires t_fail down A 2-processor failure/repair model Copyright © 2003 by K.S. Trivedi Copyright 25 Reachability Analysis t_fail 2,0 t_fail 1,1 t_repair 0,2 t_repair Reachability graph (RG) Copyright © 2003 by K.S. Trivedi Copyright 26 Example of PN Copyright © 2003 by K.S. Trivedi Copyright 27 Concurrency (or Parallelism) Copyright © 2003 by K.S. Trivedi Copyright 28 Synchronization Copyright © 2003 by K.S. Trivedi Copyright 29 Limited Resources Copyright © 2003 by K.S. Trivedi Copyright 30 Producer/consumer Copyright © 2003 by K.S. Trivedi Copyright 31 Producer/consumer with buffer Copyright © 2003 by K.S. Trivedi Copyright 32 Mutual exclusion Copyright © 2003 by K.S. Trivedi Copyright 33 Reachability Analysis • A marking is reachable from another marking if there exists a sequence of transition firings starting from the original marking that result in the new marking • The reachability set of a PN is the set of all markings that are reachable from its initial marking initial Copyright © 2003 by K.S. Trivedi Copyright 34 Reachability Analysis • A reachability graph is a directed graph whose directed nodes are the markings in the reachability set, with directed arcs between the markings representing the marking-to-marking transitions • The directed arcs are labeled with the corresponding transition whose firing results in a change of the marking from the original marking to the new marking Copyright © 2003 by K.S. Trivedi Copyright 35 Generation of the reachability graph Copyright © 2003 by K.S. Trivedi Copyright 36 Extensions of PN models arc multiplicity inhibitor arcs priority levels enabling functions (guards) Copyright © 2003 by K.S. Trivedi Copyright 37 Petri Net: Arc Multiplicity • An arc cardinality (or multiplicity) may be associated with input and output arcs, whereby the enabling and firing rules are changed as follows: – Each input place must contain at least as many tokens as the cardinality of the corresponding input arc. m p – When the transition fires, it removes as many tokens from each input place as the cardinality of the corresponding input arc, and deposits as many tokens in each output places as the cardinality of the corresponding output arc. Copyright © 2003 by K.S. Trivedi Copyright 38 Petri Net : Inhibitor Arc pi tk pj Inhibitor arcs are represented with a circle-headed arc. The transition can fire iff the inhibitor place does not contain any tokens. Copyright © 2003 by K.S. Trivedi Copyright 39 Petri Net : Inhibitor Arc Copyright © 2003 by K.S. Trivedi Copyright 40 Petri Net : Multiple Inhibitor Arc • A multiple inhibitor arc drawn from a place to a transition means that the transition cannot fire if the corresponding inhibitor place contains at least as many tokens as the cardinality of the corresponding inhibitor arc n m p • Inhibitor arcs are represented graphically as an arc ending in a small circle at the transition instead of an arrowhead Copyright © 2003 by K.S. Trivedi Copyright 41 An Example: Before or cardinality of an output arc Copyright © 2003 by K.S. Trivedi Copyright 42 An Example: After or cardinality of the output arc Copyright © 2003 by K.S. Trivedi Copyright 43 Priority levels A priority level can be attached to each PN transition. The standard execution rules are modified in the sense that, among all the transitions enabled in a given marking, only those with associated highest priority level are allowed to fire. Copyright © 2003 by K.S. Trivedi Copyright 44 Enabling Functions An enabling function (or guard) is a Boolean expression composed from the PN primitives (places, trans, tokens). The enabling rule is modified so that besides the standard conditions, the enabling function must also evaluate to be true. pi tk pj (tk) = #P1<2 & #P2=0 Copyright © 2003 by K.S. Trivedi Copyright 45 Stochastic Petri Nets (SPN) • Petri nets are extended by associating time with the firing of transitions, resulting in timed Petri nets. • A special case of timed Petri nets is stochastic Petri net (SPN) where the firing times are considered to be random variables. Copyright © 2003 by K.S. Trivedi Copyright 46 Stochastic Petri Nets (SPN) • The marking process is mapped into a continuous time Markov chain (CTMC) with state space isomorphic to the reachability graph of the PN. Copyright © 2003 by K.S. Trivedi Copyright 47 SPN: A Simple Example t1 . Server Failure/Repair t1 λ p1 p2 µ t2 p1 t2 λ Reachability graph µ λ 01 t2 . p2 CTMC t1 10 λ 10 Copyright © 2003 by K.S. Trivedi Copyright 01 µ 48 From SPN to CTMC: A Simple Example Copyright © 2003 by K.S. Trivedi Copyright 49 SPN: Poisson Process PP with rate λ λ SPN model RG = CTMC 0 λ λ 1 λ 2 ....... Copyright © 2003 by K.S. Trivedi Copyright 50 SPN: M/M/1 Queue µ λ M/M/1 RG = CTMC µ λ SPN model λ 0 λ 1 µ λ 2 ....... µ Copyright © 2003 by K.S. Trivedi Copyright µ 51 SPN: M/M/1/n Queue (1) µ λ M/M/1/n n n SPN model µ λ RG = CTMC λ 0 λ 1 µ λ 2 ....... µ Copyright © 2003 by K.S. Trivedi Copyright n µ 52 SPN: M/M/1/n Queue (2) µ λ M/M/1/n n λ µ SPN model RG = CTMC n λ 0 λ 1 µ λ 2 ....... µ Copyright © 2003 by K.S. Trivedi Copyright n µ 53 Marking dependent firing rate • A firing rate is associated with each transition. • Firing rate of a transition may be marking dependent. T n #λ Actual firing rate of T = nλ Copyright © 2003 by K.S. Trivedi Copyright 54 SPN: M/M/n/n Queue µ λ M/M/n/n µ n µ n SPN model λ # µ The use of marking-dependent firing rate Copyright © 2003 by K.S. Trivedi Copyright 55 Generalized SPN • Sometimes when some events take extremely small time to occur, it is useful to model them as instantaneous activities • SPN models were extended to allow for such modeling by allowing some transitions, called immediate transitions, to have zero firing times • The remaining transitions, called timed transitions, have exponentially distributed firing times Copyright © 2003 by K.S. Trivedi Copyright 56 Generalized SPN • The enabling rules are modified: if both an immediate and a timed transition are enabled in a marking, immediate transition has higher priority. T t Immediate transition t is enabled! • If more than one immediate transition is enabled in a marking, then the conflict is resolved by assigning firing probabilities to the immediate transitions. p1 p2 t1 t2 Transition t1 & t2 will fire with p1 and p2. Copyright © 2003 by K.S. Trivedi Copyright 57 GSPN Properties Markings (states) enabling immediate transitions are passed through in 0 time and are called vanishing. Markings (states) enabling timed transitions only, are called tangible. Since the process spends zero time in vanishing markings they do not contribute to the timed behaviour of the system and can be eliminated. Copyright © 2003 by K.S. Trivedi Copyright 58 GSPN Properties The resulting reachability graph, referred to as the Extended Reachability Graph (ERG), contains vanishing marking, and is no longer a CTMC! Need to eliminate the vanishing markings to obtain the underlying CTMC. Copyright © 2003 by K.S. Trivedi Copyright 59 Elimination of vanishing markings Situation 1 Only timed transitions are enabled. Copyright © 2003 by K.S. Trivedi Copyright 60 Elimination of vanishing markings Situation 2 One immediate and several timed transitions are enabled. ERG Copyright © 2003 by K.S. Trivedi Copyright CTMC 61 Elimination of vanishing markings Situation 3 Several immediate transitions are enabled. ERG Copyright © 2003 by K.S. Trivedi Copyright CTMC 62 Measures of Reliability & Performance Solving the model means computing the (transient/ steady state) probability vector over the state space (markings). However, the modeler wants to interact only at the PN level: the numerical procedure must be completely transparent to the analyst. There is a need to define the output measures at the PN level, in terms of the PN primitives. Copyright © 2003 by K.S. Trivedi Copyright 63 Measures of Reliability & Performance Output measures defined at the PN level. Probability of a given condition on the PN; Probability Time spent in a marking; Time Mean (first) passage time; Mean pmf of tokens in a place; pmf Expected number of firings of a PN trans Expected (throughput). All these measures can be reformulated in terms of reward functions (MRM) Copyright © 2003 by K.S. Trivedi Copyright 64 Solving models with SPN The use of SPN requires only the topology of the PN, the firing rates (or of the CDF in the general case) of the transitions and the specification of the output measures. All the subsequent steps, which consist of: generation of the reachability graph generation generation of the associated Markov chain; generation transient and s.s. solution of the Markov chain; transient computation of the relevant process measures. computation must be completely automated by a computer program, thus making the associated mathematics transparent to the user. Copyright © 2003 by K.S. Trivedi Copyright 65 GSPN Example: M/M/i/n Queue Pserver i n-i Tarrival Tservice tquick λ Pqueue Pservice Copyright © 2003 by K.S. Trivedi Copyright # µ 66 ERG for M/M/i/n Queue i=0: λ 0,0,0 λ 1,0,0 λ 2,0,0 ....... i>0 (ERG): Tarrival tquick 0,0,i 1,0,0 n-i,i,0 n-i,i-1,1 Tservice Tservice i>0 (CTMC): 0,0,i λ µ 0,1,i-1 0,1,i-1 tquick λ 2µ ... λ iµ ....... ... Tservice Tarrival 0,i,0 tquick iµ n,0,0 tquick Tarrival λ λ 1,i-1,1 0,i,0 1,i,0 1,i-1,1 1,i,0 Copyright © 2003 by K.S. Trivedi Copyright tquick Tservice λ iµ ... λ iµ n-i,i,0 67 Example: Multiprocessor with failure • Number of processors: n • Single repair facility is shared by all processors • A reconfiguration is needed after a covered fault • A reboot is required after an uncovered fault Copyright © 2003 by K.S. Trivedi Copyright 68 Assumptions: γ • The failure rate of each processor is γ • The repair times are exponentially distributed with mean 1/τ 1/ • A processor fault is covered with probability c • The reconfiguration times and the reboot times are exponentially distributed with mean 1/δ and 1/β, respectively Copyright © 2003 by K.S. Trivedi Copyright 69 GSPN Model for Multiprocessor GSPN Model of a Multiprocessor Copyright © 2003 by K.S. Trivedi Copyright 70 ERG for Multiprocessor Model (n=2) 2,0,0,0,0 Tfail tcov 1,1,0,0,0 Tuncov 1,0,0,1,0 Trecon 1,0,1,0,0 Treboot Trep 0,0,0,0,2 tquick 1,0,0,0,1 Trep Tfail 0,1,0,0,1 Extended Reachability Graph for Multiprocessor model 2,0,0,0,0 γ(1-c) 1,0,0,1,0 γc τ β 1,0,1,0,0 δ 1,0,0,0,1 γ τ 0,0,0,0,2 Reduced ERG for Multiprocessor model Copyright © 2003 by K.S. Trivedi Copyright 71 Stochastic Reward Net (SRN) • Introduced by Ciardo, Muppala and Trivedi [1989] • Structural characteristics – Extensive Marking dependency allowed for firing rates and firing probabilities – Transition Priorities – Guards (Enabling functions) for Transitions – Variable cardinality arcs Copyright © 2003 by K.S. Trivedi Copyright 72 Stochastic Reward Net (SRN) Stochastic • Stochastic characteristics – Allow definition of reward rates in terms of net level entities – Automatically generate the reward rates for the markings – Enables computation of required measures of interest Copyright © 2003 by K.S. Trivedi Copyright 73 Example: Reward Rates for Multiprocessor Availability • Reward rate at the net level for steady – state availability 1, ri = 0, # Pup ≥ 1 and (# Pcov + # Pun cov ) = 0 otherwise • Reward rate at the CTMC level for steadystate availability (n=2) 1, i = (2,0,0,0,0), (1,0,0,0,1) ri = 0, otherwise Copyright © 2003 by K.S. Trivedi Copyright 74 Analysis Procedure of SRN Stochastic Reward Nets Reachability Analysis Extended Reachability Graphs Eliminates vanishing markings Markov Reward Model Solve MRM (transient or steady-state) Measures of Interest Copyright © 2003 by K.S. Trivedi Copyright 75 SRN Summary Place Timed Transition Immediate Transition Input Arc with Multiplicity An SRN Output Arc with Multiplicity Inhibit Arc Copyright © 2003 by K.S. Trivedi Copyright 76 SRN Analysis: Step-1 Abstract the system -> SRN Model Specify in SRN Tools Finite Buffer n m m λ mµ Poisson Arrival Single Server m-stage Erlang Service time SRN of M/Em/1/n Queue Copyright © 2003 by K.S. Trivedi Copyright 77 SRN Analysis: Step-2 Reachability Analysis: Automatically Generate ERG SRN Specification Extended Reachabilty Graph Vanishing Marking Copyright © 2003 by K.S. Trivedi Copyright Tangible Marking 78 SRN Analysis: Step-3 Reachability Analysis: Automatically Generate RG Extended Reachabilty Graph Eliminate Vanishing Marking CTMC = RG Copyright © 2003 by K.S. Trivedi Copyright 79 SRN Analysis: Step-4 Solve CTMC Steady-state Analysis: A System of Linear Equations Gauss-Seidel, SOR (Successive over-relaxation) Power method, etc. Transient Analysis: A coupled system of ODE Classical ODE Methods Randomization (or Uniformization), etc. Copyright © 2003 by K.S. Trivedi Copyright 80 SRN Analysis: Step-5 Compute measures of interest Measures of interests: Blocking/Dropping Probability, Throughput, Utilization, Delay etc. Measures can be defined as reward functions which specify reward rates on net-level entities. Steps 1-5: The SPN Tool does it all! Copyright © 2003 by K.S. Trivedi Copyright 81 Example: Wireless Cellular System Channels pool (N) New Calls Call completion Handoff Calls Handoff out From neighboring cells Guard channels (g) for Handoff Calls Traffic in a cell Copyright © 2003 by K.S. Trivedi Copyright 82 Performance Measures: Loss formulas or probabilities • When a new call (NC) is attempted in an cell covered by a base station (BS), the NC is connected if an idle channel is available in the cell. Otherwise, the call is blocked blocked • If an idle channel exists in the target cell, the handoff call (HC) continues nearly transparently to the user. Otherwise, the HC is dropped dropped • Loss Formulas – New call blocking probability, Pb : Percentage of new calls rejected – Handoff call dropping probability, Pd : Percentage of calls forcefully terminated while crossing cells Copyright © 2003 by K.S. Trivedi Copyright 83 Guard Channel Scheme Handoff dropping less desirable than new call blocking! Handoff call has Higher Priority: Guard Channel Scheme GCS: g channels are reserved for handoff calls. g trade-off between Pb Copyright © 2003 by K.S. Trivedi Copyright & Pd 84 Assumptions: • λ1 : The arrival rate of new calls • λ2 : The arrival rate of hand-off calls into the • • • • cell µ1 : Service completion rate of on going calls (new or hand-off) µ2 : Service rate of hand-off outgoing calls from the cell C : Total number of channels Total g : Number of guard channels Copyright © 2003 by K.S. Trivedi Copyright 85 Description of the problem: • A hand-off call is accepted if at least one idle • • channel is available, otherwise it is dropped. A new call is accepted, if at least g+1 idle channels are available, otherwise it is blocked. Given our assumptions, the underlying traffic (performance) model is a homogeneous continuous time Markov chain of the birthdeath type. We start with an SPN Model. Copyright © 2003 by K.S. Trivedi Copyright 86 Modeling for cellular network with hard handoff Idle-channels C g+1 µ1 g λ1 New calls Talking-channels # Call-completion # λ2 Handoff-in µ2 Handoff-out G. Haring, R. Marie, R. Puigjaner and K. S. Trivedi, Loss formulae and their optimization for cellular networks, IEEE Trans. on Vehicular Technology, 50(3), 664-673, May 2001. Stochastic Petri Net Model of wireless hard handoff Copyright © 2003 by K.S. Trivedi Copyright 87 Reward Rates for the cellular network model • Reward rate for steady–state handoff call dropping probability 1, ri = 0, # Idle − channels = 0 otherwise • Reward rate for steady-state new call blocking probability 1, # Talking-channels ≥ C − g ri = 0, otherwise Copyright © 2003 by K.S. Trivedi Copyright 88 Pure Performance model • Markov Model – – – State index indicates the number of channels in use Steady-state call blocking probability (Pb) Steady-state call dropping probability (Pd) λ1+ λ2 0 1 µ1 +µ2 λ1+ λ2 ... C-g-1 (C-g)(µ1 +µ2) Pd = π C Pb = π C − g + ... + π C C-g λ2 C-g+1... C-1 (C+1-g)(µ1 +µ2) λ2 C C(µ1 +µ2) Call blocking probability Copyright © 2003 by K.S. Trivedi Copyright 89 Reward Rates for the CTMC model • Reward rate for steady–state handoff call dropping probability 1, ri = 0, i=C otherwise • Reward rate for steady-state new call blocking probability 1, i ≥ C − g ri = 0, otherwise Copyright © 2003 by K.S. Trivedi Copyright 90 Loss formulas for wireless network with hard handoff Dropping probability for handoff: Blocking probability of new calls: Note: if g=0 (no guard channel), the above expressions reduces to the classical Erlang-B loss formula Copyright © 2003 by K.S. Trivedi Copyright 91 RF Channel Failure/Recovery A talking channel may fail! First we discuss a pure availability model Then we discuss a composite performance/availability model Then we discuss call recovery Copyright © 2003 by K.S. Trivedi Copyright 92 Pure Availability model: • Markov Model: (C channels total) – – – – – Each channel has MTTF and MTTR Steady-state system Unavailability : U Steady-state system Availability : A Instantaneous system Availability : At Downtime : downtime (in minutes) C/MTTF C C-1 1/MTTR 1/MTTF (C-1)/MTTF C-2 ... 1/MTTR Copyright © 2003 by K.S. Trivedi Copyright 1 1/MTTR 0 93 SPN Availability model # C 1/MTTF Repair Avail_channels 1/MTTR Copyright © 2003 by K.S. Trivedi Copyright 94 Composite SPN Model Handoff_in_call New_call λ2 λ1 Completion_of_call # In_use g+1 g µ1 # µ2 # C 1/MTTF Repair Avail_channels Handoff_out_call 1/MTTR # 1/MTTF Copyright © 2003 by K.S. Trivedi Copyright 95 RF Channel Recovery A talking channel may fail! APS (Automatic Protection Switch) Widely used in ATM Automatically switched to an idle channel if available. Otherwise, queued until an idle channel is available. Copyright © 2003 by K.S. Trivedi Copyright 96 SRN Model of Channel Recovery λ2 λ1 µ1 µ2 µ1 Copyright © 2003 by K.S. Trivedi Copyright 97 Size of State Space Model Tangible Markings Vanishing Markings Times Transitions Recovery 440 381 2400 State space for C = 20 (total number of channels) Copyright © 2003 by K.S. Trivedi Copyright 98 WFS Example Revisited as SPNs Copyright © 2003 by K.S. Trivedi Copyright 99 WFS example first SPN 2 1 Pwsup Pfsup µw # λw Pwsrep Tfsrep Tfsfail Twsfail λf µf Pfsdn Pwsdn Copyright © 2003 by K.S. Trivedi Copyright 100 Reachability Graph of the SRN Pwsup Pfsup Pwsdn Pfsdn 2100 2001 1110 1011 0120 0021 Copyright © 2003 by K.S. Trivedi Copyright 101 GSPN Model for WFS Example • Assume that when a workstation fails, with probability c the failure is properly detected the • With the probability (1 – c) the failure is not detected, leading to the corruption and the failure of the file-server Copyright © 2003 by K.S. Trivedi Copyright 102 GSPN Model for WFS Example with imperfect coverage Pwsup Pwsfl Pfsup # Tfsfl Pwst Tfsrp Twsrp twscv twsuc Pwsdn Pfsdn Copyright © 2003 by K.S. Trivedi Copyright 103 Copyright © 2003 by K.S. Trivedi Copyright 104 Reachability Graph of the GSPN wsrp 21000 fsfl wsfl 11100 wsfl wscv wsuc fsfl fsrp 20001 wsrp 10101 wscv 11010 wsfl 01110 wscv wsuc fsfl fsrp 10011 wsfl 00111 wscv Copyright © 2003 by K.S. Trivedi Copyright 01020 fsrp 00021 105 The CTMC for the example µw 21000 λf µf 20001 µw 2λwc λwc 2λw(1-c) 2λ w 11010 λf µf λw(1-c) 10011 Copyright © 2003 by K.S. Trivedi Copyright λw 01020 λf µf 00021 106 System Availability for the Example Copyright © 2003 by K.S. Trivedi Copyright 107 GSPN Model for WFS Example with non preemptive repair priority Pwsup Pfsup Twsfl Twsrp Tfsfl Tfsrp Pwsdn Pfsdn Pwsrp Pfsrp t1 t0 Prepair Copyright © 2003 by K.S. Trivedi Copyright 108 CTMC for the WFS example with non preemptive repair priority λf µf 1 µf 2λw µw λf 2 4 3 λf 7 λw λf 2λw µw µw λw 6 λw µw 5 8 µf Copyright © 2003 by K.S. Trivedi Copyright 109 SPN Software Packages SPNP v6 (Duke, USA) GreatSPN (Torino, Italy) DSPNexpress (Dortmund, Germany) TimeNet (Hamburg, Germany) etc. Copyright © 2003 by K.S. Trivedi Copyright 110 SPNP Software Package Copyright © 2003 by K.S. Trivedi Copyright 111 SPNP • • • • • • • • Installed at over 250 Sites; companies & universities Ported to Most Architectures and Operating Systems Used For Performance, Dependability and Performability Steady-State as well as Transient Analysis Analytic-numeric methods for Markovian models. Simulation for non-Markovian and fluid models Written in C Language GUI now available Copyright © 2003 by K.S. Trivedi Copyright 112 SOME INDUSTRIAL USES • HP – Cluster Availability Modeling – Server Availability – Mass Storage Arrays Availability Modeling • MOTOROLA – Recovery strategies in wireless handoff: – Proposed and modeled several strategies – Fixed-point iteration used – Software rejuvenation in CMTS • IBM – Software rejuvenation for a cluster system • Boeing, EMC, … … Copyright © 2003 by K.S. Trivedi Copyright 113 SPN CHARACTERISTICS • Structural characteristics: – – – – – Marking dependency Priorities Guards Resampling policies, for general distributions Variable cardinality arcs • Stochastic characteristics: – Allow definition of reward rates in terms of net level entities – Automatically generate the reward rates for the markings Copyright © 2003 by K.S. Trivedi Copyright 114 STEPS IN ANALYTIC NUMERIC SOLUTION OF MARKOVIAN SRN Stochastic Reward Net Generates all markings of the SRN by considering all the enabled transitions in each marking. Classify the markings as Tangible and Vanishing markings. Extended Reachability Graph Eliminates the vanishing markings Markov Reward Model Solve MRM for Steady-State Transient behavior using known methods Copyright © 2003 by K.S. Trivedi Copyright 115 DISCRETE EVENT SIMULATION ANALYSIS • Can be used for: • • • – Markovian SRN – non-Markovian SRN – Fluid SPN FSPN (Fluid Stochastic Petri net) Used as a model for: – Systems involving fluid variables – Approx. of models with a large number of tokens No need to generate the reachability graph Possibility to give the number of replications or the desired relative error. Copyright © 2003 by K.S. Trivedi Copyright 116 DISTRIBUTIONS AVAILABLE FOR SIMULATION – Exponential – Constant (including Immediate) – Uniform – Truncated normal – Weibull – Lognormal – Geometric – Erlang – Pareto – Cauchy – Beta – Gamma – Poisson Copyright © 2003 by K.S. Trivedi Copyright 117 Solution Technique in SPNP Copyright © 2003 by K.S. Trivedi Copyright 118 Plots can be combined to be compared SRN example Copyright © 2003 by K.S. Trivedi Copyright 119 Recent Research on SPN Efficient Solution Techniques Largeness Stiffness Extension to non-Markovian SPNs Markov Regenerative Stochastic Petri nets (MRSPN); solution methods Fluid stochastic Petri nets (FSPN); solution methods for first order & second order including boundary conditions Copyright © 2003 by K.S. Trivedi Copyright 120 Research on Simulation Methods • Simulation methods – Implementation of other general distributions – Importance sampling (for rare events) – Sensitivity analysis – Non-linear FSPNs – 2nd order FSPNs Copyright © 2003 by K.S. Trivedi Copyright 121 Largeness Problem: Model complexity State space explosion Solution: Decomposition Kronecker algebra Fixed-point iteration Copyright © 2003 by K.S. Trivedi Copyright 122 Stiffness Problem: Parameters on different time scales ATM example Cell level Burst level ns ms Connection level minutes Ill-conditioned matrices Solution: Decomposition Fixed-point iteration Stiffly stable numerical methods (e.g., Uniformization, implicit methods) Copyright © 2003 by K.S. Trivedi Copyright time 123 Non-Markovian SPN Transition Firing Time: not exponentially distributed H.Choi,V. Kulkarni, K. Trivedi Markov regenerative stochastic Petri net (MRSPN) Performance Evaluation, 20, 337-357, 1994 (A special case: At most one general transition can be enabled in any marking). A. Bobbio and A. Puliafito and M. Telek and K. Trivedi. Recent developments in non-Markovian stochastic Petri nets. Journal of Systems Circuits and Computers, 8:1, 119-158, 1998. Copyright © 2003 by K.S. Trivedi Copyright 124 Fluid Petri Net Fluid stochastic Petri net (FSPN) Introduced by K. Trivedi and V. Kulkarni (1993) Allow both discrete and continuous places Useful in fluid approximation of discrete queuing system Powerful formalism of stochastic fluid queueing networks Boundary conditions complicated. Solution techniques under investigation. Copyright © 2003 by K.S. Trivedi Copyright 125 The Fluid Petri Net Model FPN is an extension of PN that is able to model the coexistence of discrete and continuous variables. The primitives of FPN (places, transitions and arcs) are partitioned in two groups: discrete primitives that handle discrete tokens (as in standard PN); continuous (or fluid) primitives that handle continuous (fluid) quantities. fluid arcs are assigned instantaneous flow rates. Copyright © 2003 by K.S. Trivedi Copyright 126 Fluid Petri Nets Copyright © 2003 by K.S. Trivedi Copyright 127 References • http://www.ee.duke.edu/~kst/ then click on Stochastic Petri Nets • K. Trivedi, Probability and Statistics with Probability Reliability, Queuing, and Computer Science Applications, 2nd Ed., John Wiley and Sons, New York, 2001 Copyright © 2003 by K.S. Trivedi Copyright 128 ...
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This note was uploaded on 04/08/2010 for the course COMPUTER E 409232 taught by Professor Mohammadabdolahiazgomiph.d during the Spring '10 term at Islamic University.

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