Lecturer: Olga Galinina
|
|
- Robyn Stephens
- 5 years ago
- Views:
Transcription
1 Lecturer: Olga Galinina
2 Outline Motivation Modulated models; Continuous Markov models Markov modulated models; Batch Markovian arrival process; Markovian arrival process; Markov modulated arrival process; Switched Markov modulated arrival process. Discrete Markov models Discrete Markov modulated models; Discrete-time batch Markovian arrival process; Discrete-time Markovian arrival process; Markov modulated Bernoulli process; Switched Markov modulated Bernoulli process; Discrete-time switched Poisson process; Discrete-time switched Bernoulli process; Fitting parameters example.
3 Why non-renewal models? Classic renewal processes: strictly stationary; completely uncorrelated; single arrivals. In general inadequate to capture traffic properties Real traffic: may not be strictly stationary; may be correlated; multiple arrivals are allowed. Note! We have to be close to reality! 3
4 Modulated models Basic concept: one process X i (t) determines the state S(t); states are associated with other processes. Figure 1: Graphical representation of general modulated process. What is important about this process: values of {S(t), t {T}} are not observable; values of {X i (t), t {T}}, i = 1,,, M are observable. 4
5 Modulated models In general: no restrictions on the choice of {S(t), t {T}} ; no restrictions on the choice of {X i (t), t {T}}, i = 1,,, M. Why these processes are non-renewal: value of the process depends on the state; in special case there is autocorrelation. We are interested in special case: modulating process is Markovian; processes in states are renewal. These are called Markov modulated processes 5
6 Markov modulated processes We distinguish between: continuous-time model interarrival times; discrete-time model number of arrivals in slots. Classification is related to measurements: continuous interarrival times time-consuming; discrete number of arrivals in slots easier to do. Figure : Discrete and continuous measurements. 6
7 Continuous-time time Markov models 7
8 Batch Markovian AP (BMAP) To define BMAP we start considering Poisson process assuming: no arrivals occurred prior to t=0; we consider it in terms of arrivals prior to time t: N(t); rate of the process is assumed to be λ. {N(t),t 0}, N(t) {0,1, } can be considered as pure birth process: states N(t) {0,1, } denote number of arrivals; exponential sojourn time in state i; after that process jumps to state i+1 resulting in exponential interarrival time. Figure 3: Interpretation of Poisson arrival process as pure birth process. 8
9 Batch Markovian AP (BMAP) Infinitesimal generator of {N(t), t 0}: d 0 d d0 d1... Q =, 0 0 d0... d1 = λ, d0 = d1 = λ. If the process is batch Poisson we have: d0 d1 d d3 0 d0 d1 d Q =, 0 0 d0 d1 d = λ p, i = 1,,...; d = d = λ. i i 0 i i=1= 1 Here p i is the probability that size of the batch is i. 9
10 Batch Markovian AP (BMAP) Batch Markovian arrival process: extension of the batch Poisson: interarrival times are no longer exponential; Markovian structure is still preserved. Consider process {N(t),J(t), t 0}, N(t) {0,1, }, J(t) {0,1,,M}: ( 0) ( 1) ( ) ( 3) 0 D ( 0 ) D ( 1 ) D ( ) 0 0 D( 0) D( 1) D D D D Q =. D(k), k = 0, 1,... are M M matrices; D(0): negative diagonal elements and non-negative off-diagonal; D(k), k = 1,,... are non-negative: D = i=0= 0 ( ) D i. 10
11 Batch Markovian AP (BMAP) Important notes about BMAP: most general analytically tractable continuous-time Markov modulated process; allows arbitrary distribution of the interarrival times; ACF: sum of exponential terms. Constructive interpretation of BMAP: continuous-time Markov chain {S(t), t R}, S t {1,,, M}: sojourn time is exponentially distributed with parameters λ i, i=1,,, M; when MC changes its state from i to j a batch of arrivals is generated. Denote probabilities of transitions with k arrivals as: p ij (k), i=1,,, M. we assume p ii (0)=0, i=1,,. 11
12 Batch Markovian AP (BMAP) How to characterize D-BMAP: number of states, M; intensitiesout of states λ i, i=1,,..., M; probabilitiesp ij (k), i, j=1,,..., M, k = 0, 1, How to have this information in compact form: matrices D(k) where each element is: ( ) = λ,, = 1,,...,, = 1,,... d k p i j M k ij i ij matrix D(0) each element of which is λi pij, i j dij =, λ i, i = j where for infinitesimal generator D the following holds: D D( i). = i=1 1
13 Batch Markovian AP (BMAP) Matrices D(0), D(1),..., D(k),... take the form: λ 1 p1 ( 0 ) λ1 p13 ( 0 ) λ1... p1m ( 0 ) λ1 p1( 0) λ λ p3 ( 0 ) λ... pm ( 0) λ D( 0 ) =, pm1( 0) λm pm ( 0) λm pm3( 0 ) λm... λ M p 11( 1) λ 1 p 1 ( 1) λ 1 p 13( 01 ) λ 1... p 1 M ( 1) λ 1 p1( 1) λ p ( 1) λ p3 ( 1 ) λ... pm ( 1) λ D( 1 ) =, pm1( 1) λm pm ( 1) λm pm3( 1 ) λm... pmm ( 1) λ M ( ) D k ( ) ( ) ( 01 )... M ( ) ( ) ( ) ( 1 )... ( ) p 11 k λ 1 p 1 k λ 1 p 13 λ 1 p 1 k λ 1 p1 k λ p k λ p3 λ pm k λ = ( ) ( ) ( )... pm k λm pm k λm pm λm pmm ( k) λ M 13
14 Markovian arrival process: MAP MAP is a special case of BMAP: only single arrivals are allowed. MAP process {W(t), t R} is defined as: d 1 = λ p 1, i, j = 1,,..., M ; k = 0,1. d ( ) ( ) ( ) λi ij ( ) ij i ij ij p 1, i j 1 =. λ i, i = j D = D(0) + D(1). Mean is given by: [ ] 1 ( ( 0 )) 1 E W = = π D e λ λ is the overall intensity of the process. 14
15 Markov modulated arrival process: MMPP MMPPis a special case of MAP: arrivals are allowed when state changes from i to i, i=1,,,m. MMPP process is defined as: λ i pij ( 1 ), i = j d ij ( 1 ) =, 0, i j D = D(0) + D(1). In matrix form: d ij ( ) ( ) λi pij 1, i j 0 =. λi, i = j ( ) ( ) ( ) D 0 = D Λ, D 1 = Λ, D k = 0, k 1. Λ = diag ( λ are rates of Poisson process in 1, λ,..., λm ) = diag ( λ ), λi states i = 1,,, M; λ, D full description is then given by ( ) 15
16 Markov modulated arrival process: MMPP Characteristics of MMPP: steady-state vector π = (π 1, π,..., π M ) of modulating CTMC: π D = 0, π e = 1. mean arrival rate is given by: θ T λ = π ( ) ( 1 ) kd k e = π D e = πλ e = πλ. k = 1 if CTMC is irreducible aperiodic, autocovariance function is given by: N 1 1 i CW ( τ ) = E W ( t) W ( t + τ ) = λδ ( τ ) + ϕ0 + ϕ ie γ τ, ( ) = 1, τ = 0. ( ) = 0, τ 0 δ τ δ τ γ, i= 0, 1,, N 1 are N 1 eigenvaluesof CTMC given that ; i γ 0 = 0 if CTMC is irreducible aperiodic all eigenvalues and real; i= 1 16
17 Markov modulated arrival process: MMPP Distributions of the MMPP: PF number of arrivals is a weighted sum of Poisson distributions: N k i π iλi e p Pr { W ( t) k} λ k = = =, k = 0,1,..., k i= 1! π i is the steady state probability of CTMC is state i; PDF of interarrival times is a weighted sum of exponentials: N iw pw ( w) = π iλi e λ, i= 1 this is known as hyperexponential distribution; recall that C > 1 and C = 1 in a limiting case. What we can model with MMPP: empirical distributions with high variability: C > 1 provides simple check of MMPP suitability! ACF exhibiting (sum of) exponential decay. 17
18 Switched Markov modulated arrival process: SMMPP SMMPP is a special case of MMPP: modulating Markov chain has only states.. SMMPP is given by: λ1 0 r, 1 r Λ = D =, 0 λ r r D ( ) 0 = D Λ. Steady-state probabilities of modulating CTMC are: π λ r λ r =, π = λ1r + λr1 λ1r + λr1 π is the vector containing these probabilities. 18
19 Switched Markov modulated arrival process: SMMPP CDF F(x) = Pr{X x} of interarrival times is given by: ( D Λ) x 1 u1x u x F ( x) = 1 π e ( Λ D) Λ e = 1 qe + ( 1 q) e, e = ( 1,1 ),0 < q < 1. Probability density function (PDF) is: A u x ( ) ( ) 1 u x f x qu e q u e q = + 1,0 < < 1. 1 Autocorrelation function is given by: ( 1 1 )( + 1 [ + 1] ) ( ) [ ] K X k = E X E X X k E X k = ( ) ( 1 ( ) ) k + 1 k π D D eπ = Λ Λ Λ Λ ( Λ D) Λ e = Aσ, k = 1,,... where A and σ are given by: ( ) r r ( + ) ( + + ) λ1 λ 1 λ1λ =, σ =. λ1r λr1 λ1λ λ1r λ r λ 1 1λ + λ1r + λr1 19
20 Switched Markov modulated arrival process: SMMPP What is important about SMMPP: distribution is hyperexponential; ACF decays exponentially. What we may capture by SMMPP: interarrival distributions with: monotone decreasing behavior; coefficient of variation: C > 1. q 1 q + [ ] ( [ ] σ X E X E X ) u1 u C = = = 1 1. ( E [ X ]) ( E [ X ]) q 1 q + u1 u exponentially decaying ACF only! 0
21 Switched Markov modulated arrival process: SMMPP Figure 7: Possible distributions of SMMPP. 1
22 Switched Markov modulated arrival process: SMMPP Figure 8: Possible normalized ACFs of SMMPP.
23 Discrete Markov modulated models 3
24 Discrete Markov modulated models What is special about such models: modulated process is discrete Markov in nature transition probability matrix is in the form: d d d M d d d, 1, 1,,...,. M 1 M Q = d ji = j = M i= 1 d d d M1 M MM What is interesting: usually easier to deal with; may have arbitrary distribution; may have complex ACF structure: sum of geometrical terms. 4
25 Discrete Markov modulated models Discrete processes: time is divided into intervals (slotted); durations of intervals are the same t; some arrivals may occur in each interval. Figure 11: Illustration of discreteness of the arrival process. Note the following: interpretation of discrete processes can be considered as approximations of real arrivals; sometimes this is a natural way. we are going to work with number representation; intervals representation is also possible. 5
26 Discrete-time time batch Markovian arrival process Basic characteristics: most general analytically tractable discrete Markov modulated process; allows arbitrary distribution of the number of arrivals in a slot; ACF: sum of geometrical terms. Assume time axis is slotted; the slot duration is constant and given by t = (t i+1 t i ); discrete-time homogenous aperiodic, irreducible MC {S(n), n=0, 1,...}: state spaces(n) { 1,,..., M } ; transition probability matrix D. {W(n), n=0, 1, } is D-BMAP with MMC is {S(n), n=0, 1,...} if: value of {W(n), n=0, 1, } is function of the current state of {S(n), n=0, 1,...} 6
27 Discrete-time time batch Markovian arrival process How to completely define D-BMAP: matrices D(k), k = 0, 1, : state change from ito j, i,j = 0, 1,... ; arrival of k customers. Example: d ij (0): transition from state ito state j without any arrivals; d ij (k): transition from state ito state j with a batch arrival of size k. In general for d ij (k) we have: d ij (k) = Pr{W(n) = k, S(n) = j S(n 1) = i}, k= 0, 1,... Note the following: for pair (i, j), d ij (k), k = 0, 1,... are called conditional probability functions: M j= 1 k= 0 ( ) d k = 1, j = 0,1,..., M. ij For different pairs (i, j) d ij (k) are allowed to be different. 7
28 Discrete-time time batch Markovian arrival process Figure 1: Illustration of the D-BMAP. 8
29 Discrete-time time batch Markovian arrival process π = π π π ( 1,,..., M) Let be the vector of stationary probabilities of {S(n)}: i ( n) π = lim π, i = 1,,... n We can find π = π, π,..., π using: i ( 1 M) π D = π. π e = 1 easiest way to compute: D i and take any row where i is large (> 1000): Example of how to compute π : D = = Hence, π = , ( ) 1000, D. 9
30 Discrete-time time batch Markovian arrival process Using the mean arrival rate in the slot is: E W = kd k e, e = 1,...,1. π [ ] ( ) ( ) π k= 1 The variance of D-BMAP is: D W R 0 k D k e E W. ( ) [ ] = ( ) = ( ) [ ] W π k=1= 1 Let R W (i), i0, 1,... be the ACF of the D-BMAP: ( ) ( ) ( ) [ ] ( ) W ( ) [ ] RW i = E W n W n + 1 E W, i 0, R 0 = D W. The ACFof D-BMAP is: R i = kd k D kd k e E W, i 0. W π k= 1 k= 1 ( ) i 1 ( ) ( ) ( ) [ ] 30
31 Discrete-time time batch Markovian arrival process The mean process of D-BMAP: {G(n), n=0, 1,...} with G(n) = G i : M j= 1 k= 1 ( ) G = kd k, i = 1,,..., M. i G = G,G,..., G M ( ) 1 ij is the mean vector of D-BMAP. The mean input rate in the slot is given by: M [ ] = π = [ ] E G G E W k= 1 The variance of the mean process of D-BMAP is given by: i i. M ( ) [ ] = ( ) = π [ ] D G R 0 G E G. G i i k= 1 The ACF of the mean process is given by: ( ) i 1 ( ) R ( ) ( [ ]) G i = π kd k D kd k e E G, i 0. k= 1 k= 1 31
32 Discrete-time time batch Markovian arrival process Advantages of using D-BMAP: quite general process; analytically tractable. Shortcomings of using D-BMAP: really hard to parameterize: we have to estimate matrices D(k), k = 0, 1, ; M M k max parameters. We usually use a special case of D-BMAP: D-MAP: discrete-time Markovian arrival process; D-MMBP: discrete-time Markov modulated batch process; D-SBP: discrete-time switched batch process; D-SPP: discrete-time switched Poisson process; D-SBP: discrete-time switched Bernoulli process. 3
33 Discrete-time time Markovian arrival process D-MAP is a special case of D-BMAP: only single arrival in a slot is possible D(0) and D(1)! Mean is given by: E W = πd 1 e, [ ] ( ) The variance of D-MAP is: The ACF of D-BMAP is: D W = R 0 = πd 1 e- E W, ( ) [ ] ( ) ( ) [ ] W 1 R ( i ) π D( 1) D i W = e- E[ W ], i 0. ( ) Note the following: D-MAP reduces versatility of D-BMAP in terms of batch arrivals; D-MAP still have different conditional PFs for each different pair of states (i, j)! 33
34 Markov modulated Bernoulli process MMBPis a special case of D-BMAP: conditional PFs depends on the current state only. Recall for D-BMAP we had: { } ( ) ( ) ( ) ( ) d ( k) = d ( k), j l. d k = Pr W n = k, S n = j S n 1 = i, k = 0,1,..., i, j = 0,1,..., M. ij in general, ij il too many conditional PFs to determine; these conditional PFs depends on pair of states (ij). ForMMBPwe have: ij ( ) = ( ),. d k d k j l il in overall, we only have M conditional PFs; now, it does not matter to which state transition occurs; ACF, meanand variancecan be obtained using the same expression as for D-BMAP 34
35 Markov modulated Bernoulli process When it does not matter to which state transition occurs we may define: M { } ( ) ( ) ( ) ( ) a k = Pr W n = k, S n = j S n 1 = i, k = 0,1,..., i, j = 0,1,..., M. i j= 1 a i (k), k = 0, 1,..., i = 1,,... are conditional PFs of arrivals; these conditional PFs depend on the state from which transition occurs. For a i (k), k = 0, 1,..., i = 1,,...: k=0 i ( ) = 1,, = 0,1,...,. a k i j M 35
36 Switched D-MMBP Special case of D-MMBP: only two states of the MMC. Assume that transition probability matrix of MMC has the following form: 1 α α D =. β 1 β Steady-state distribution expressed in terms of α and β: ACFof the mean process is: β α π1 =, π1 =. α + β α + β i G ( ) = [ ] λ, = 0,1,... R i D G i D[G] is the variance of the mean process; λ is single non-unit eigenvalueof the MMC: MMC has only two states: λ 0 = 0, 0 λ 1 < 1. 36
37 Switched D-MMBP We can express ACF of the mean process in terms of G G i i R ( ) 1 ( 1 ) [ ]( 1 ), 0,1,... G i = αβ α β D G α β i α + β = = λ = (1 α β); G 1 and G are means in states 1 and, respectively. Normalized ACF of the mean process is then: Note! ( i ) [ ] KG i KG ( i) =, i 0,1,... D G = λ = we have no simple relation between R G (i) and R W (i) except for: ( ) ( ) ( ) ( ) R i = R i, i = 0,1,..., R 0 = R 0 + x, W G W G (when conditional PFs are close to Poisson, x is close to E[W]). process may produce fair approximation for geometrically decaying ACF; conditional PFs are allowed to be arbitrary. 37
38 Discrete-time time switched Poisson process Special case of switched D-MMBP: conditional PFs are no longer arbitrary; in this particular case it is Poisson. D-SPP: 1, i = 0 R W ( i) = R G ( i) + E[ W] δ i, δ i =. 0, i = 1,,... [ ] = [ ] = π1 1 + π E W E G G G is the mean of SPP. Note! ACF has geometrical decay; distribution is a mixture of two Poisson distributions (not Poisson). 38
39 Discrete-time time switched Poisson process Figure 13: Possible behavior of the distribution of D-SPP. 39
40 Discrete-time time switched Poisson process What is required to parameterize D-SPP: transition probability matrix of MMC: α and β; means in states 1 and : G 1 and G ; recall that mean completely determine Poisson distribution. Special case of D-SPP is interrupted D-SPP: mean in state 1 is zero (no arrivals); mean in state is not zero. Characteristics of interrupted D-SPP: ACF still has geometrical decay; distribution is Poisson. 40
41 Discrete-time time switched Bernoulli process Special case of switched D-MMBP: conditional PFs are no longer arbitrary; in this particular case it is Bernoulli: only single parameter in each state: probability of arrival: { } ( ) ( ) ( ) ( ) d 1 = Pr W n = k, S n = 1 S n 1 = i = p, i, j = 1,. ij note that conditional PFs depend only on the current state. i D-SBP: setting p 1 =1, p =0 (or vice versa): W ( ) = ( ), = 0,1,..., R i R i i G (you can check it by inserting mean and variance in expression for ACF of D-BMAP). when both p 1 and p are not zero this property does not hold. 41
42 Fitting parameters example 4
43 Fitting parameters. Switched Markov modulated arrival process: SMMPP Parameters of PDF can be found using λ 1, λ, r 1, r : u1 x u x f x = qu e + 1 q u e,0 < q < 1. u q ( ) ( ) 1 λ + λ + r + r δ λ + λ +, u r + r + δ = =, λ r1 + λ1 r u =. λ r + λ r u u u u ( 1 1 )( 1 ) δ can be found as follows: 1 ( r r ) 4 r r. δ = λ λ Recall: parameters A and σof ACF can be found using λ 1, λ, r 1, r : A k K X ( k ) = Aσ, k = 1,,... ( ) r r ( + ) ( + + ) λ1 λ 1 λ1λ =, σ =. λ 1 1 1r λ r1 λ1λ r r λ1r λ r λ λ + λ + λ
44 Fitting parameters SMMPPis completely defined by λ 1, λ, r 1, r ; histogram and ACF of data are completely defined by u 1, u, σ and q; there is unique SMMPP capturing empirical data. Algorithm: estimate u 1, u, q from empirical pdf; estimate σ from empirical ACF; find λ 1, λ, r 1, r using the following: 1 λ1 = q ( 1 σ )( u1 u ) σ u1 u ( q ( 1 σ )( u1 u ) σ u1 u ) 4 σ u1u , λ = r 1 1 r = = u1u ( λ1 q ( u1 u ) u ) λ1u1 λ1q ( u1 u ) u ( u λ )( u λ ) λ λ 1 ( λ u )( λ + r u ) λ λ 1,., 44
45 Fitting two first moments and NACF Determine u 1, u, q as follows: hyperexponentialdistribution with balanced means q/u 1 = (1 q)/u ; probabilities q can be found as: 1 C 1 q = 1 +. C 1 + rates u1 and u are given by: u [ ] 1 ( q) E [ Y ] q 1 =, u =. E Y Determine σ by setting σ = K Y (1) or minimizing: ( m) ( ) m= m m 0 1 K Y σ γ = m K m 0 m= 1 Y K Y (i), i= 1,,... is the normalized ACF of empirical data; m 0 is the intervals for which K Y (m 0 )
46 Example of fitting Interarrival times and corresponding statistics Figure 9: Trace, histogram and NACF of interarrival times.. 46
47 Example of fitting Moments of data: E[Y] =.7, σ [Y] = 4.978, C[Y] = What are conclusions about data: looks like data are exponential C [Y ] 1; we can still use hyperexponentialdistribution to approximate exponential. use empirical moments estimate parameters of H (q, u 1, u ) as: q = 0.53, u 1 = 0.476, u = 0.4. use empirical NACF estimate σ as σ = K(1) = determine parameters of SMMPP λ 1, λ, r 1, r as: λ 1 = 0.45, λ = 0.170, r 1 =.557E 3, r =
48 Example of fitting Figure 10: Visual comparison of empirical characteristics and characteristics of the model. 48
Lecturer: Olga Galinina
Renewal models Lecturer: Olga Galinina E-mail: olga.galinina@tut.fi Outline Reminder. Exponential models definition of renewal processes exponential interval distribution Erlang distribution hyperexponential
More informationStochastic process. X, a series of random variables indexed by t
Stochastic process X, a series of random variables indexed by t X={X(t), t 0} is a continuous time stochastic process X={X(t), t=0,1, } is a discrete time stochastic process X(t) is the state at time t,
More informationIrreducibility. Irreducible. every state can be reached from every other state For any i,j, exist an m 0, such that. Absorbing state: p jj =1
Irreducibility Irreducible every state can be reached from every other state For any i,j, exist an m 0, such that i,j are communicate, if the above condition is valid Irreducible: all states are communicate
More informationRecap. Probability, stochastic processes, Markov chains. ELEC-C7210 Modeling and analysis of communication networks
Recap Probability, stochastic processes, Markov chains ELEC-C7210 Modeling and analysis of communication networks 1 Recap: Probability theory important distributions Discrete distributions Geometric distribution
More informationPart I Stochastic variables and Markov chains
Part I Stochastic variables and Markov chains Random variables describe the behaviour of a phenomenon independent of any specific sample space Distribution function (cdf, cumulative distribution function)
More informationGI/M/1 and GI/M/m queuing systems
GI/M/1 and GI/M/m queuing systems Dmitri A. Moltchanov moltchan@cs.tut.fi http://www.cs.tut.fi/kurssit/tlt-2716/ OUTLINE: GI/M/1 queuing system; Methods of analysis; Imbedded Markov chain approach; Waiting
More informationIEOR 6711, HMWK 5, Professor Sigman
IEOR 6711, HMWK 5, Professor Sigman 1. Semi-Markov processes: Consider an irreducible positive recurrent discrete-time Markov chain {X n } with transition matrix P (P i,j ), i, j S, and finite state space.
More informationMARKOV PROCESSES. Valerio Di Valerio
MARKOV PROCESSES Valerio Di Valerio Stochastic Process Definition: a stochastic process is a collection of random variables {X(t)} indexed by time t T Each X(t) X is a random variable that satisfy some
More informationOutlines. Discrete Time Markov Chain (DTMC) Continuous Time Markov Chain (CTMC)
Markov Chains (2) Outlines Discrete Time Markov Chain (DTMC) Continuous Time Markov Chain (CTMC) 2 pj ( n) denotes the pmf of the random variable p ( n) P( X j) j We will only be concerned with homogenous
More informationMarkov Chains. X(t) is a Markov Process if, for arbitrary times t 1 < t 2 <... < t k < t k+1. If X(t) is discrete-valued. If X(t) is continuous-valued
Markov Chains X(t) is a Markov Process if, for arbitrary times t 1 < t 2
More informationReadings: Finish Section 5.2
LECTURE 19 Readings: Finish Section 5.2 Lecture outline Markov Processes I Checkout counter example. Markov process: definition. -step transition probabilities. Classification of states. Example: Checkout
More informationLecture 4a: Continuous-Time Markov Chain Models
Lecture 4a: Continuous-Time Markov Chain Models Continuous-time Markov chains are stochastic processes whose time is continuous, t [0, ), but the random variables are discrete. Prominent examples of continuous-time
More informationCharacterizing the BMAP/MAP/1 Departure Process via the ETAQA Truncation 1
Characterizing the BMAP/MAP/1 Departure Process via the ETAQA Truncation 1 Qi Zhang Armin Heindl Evgenia Smirni Department of Computer Science Computer Networks and Comm Systems Department of Computer
More information(b) What is the variance of the time until the second customer arrives, starting empty, assuming that we measure time in minutes?
IEOR 3106: Introduction to Operations Research: Stochastic Models Fall 2006, Professor Whitt SOLUTIONS to Final Exam Chapters 4-7 and 10 in Ross, Tuesday, December 19, 4:10pm-7:00pm Open Book: but only
More informationData analysis and stochastic modeling
Data analysis and stochastic modeling Lecture 7 An introduction to queueing theory Guillaume Gravier guillaume.gravier@irisa.fr with a lot of help from Paul Jensen s course http://www.me.utexas.edu/ jensen/ormm/instruction/powerpoint/or_models_09/14_queuing.ppt
More informationECE-517: Reinforcement Learning in Artificial Intelligence. Lecture 4: Discrete-Time Markov Chains
ECE-517: Reinforcement Learning in Artificial Intelligence Lecture 4: Discrete-Time Markov Chains September 1, 215 Dr. Itamar Arel College of Engineering Department of Electrical Engineering & Computer
More informationM/G/1 and M/G/1/K systems
M/G/1 and M/G/1/K systems Dmitri A. Moltchanov dmitri.moltchanov@tut.fi http://www.cs.tut.fi/kurssit/elt-53606/ OUTLINE: Description of M/G/1 system; Methods of analysis; Residual life approach; Imbedded
More informationJitter Analysis of an MMPP 2 Tagged Stream in the presence of an MMPP 2 Background Stream
Jitter Analysis of an MMPP 2 Tagged Stream in the presence of an MMPP 2 Background Stream G Geleji IBM Corp Hursley Park, Hursley, UK H Perros Department of Computer Science North Carolina State University
More informationReview. DS GA 1002 Statistical and Mathematical Models. Carlos Fernandez-Granda
Review DS GA 1002 Statistical and Mathematical Models http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall16 Carlos Fernandez-Granda Probability and statistics Probability: Framework for dealing with
More informationCDA5530: Performance Models of Computers and Networks. Chapter 3: Review of Practical
CDA5530: Performance Models of Computers and Networks Chapter 3: Review of Practical Stochastic Processes Definition Stochastic ti process X = {X(t), t T} is a collection of random variables (rvs); one
More informationIntroduction to Queuing Networks Solutions to Problem Sheet 3
Introduction to Queuing Networks Solutions to Problem Sheet 3 1. (a) The state space is the whole numbers {, 1, 2,...}. The transition rates are q i,i+1 λ for all i and q i, for all i 1 since, when a bus
More informationNon Markovian Queues (contd.)
MODULE 7: RENEWAL PROCESSES 29 Lecture 5 Non Markovian Queues (contd) For the case where the service time is constant, V ar(b) = 0, then the P-K formula for M/D/ queue reduces to L s = ρ + ρ 2 2( ρ) where
More informationLECTURE #6 BIRTH-DEATH PROCESS
LECTURE #6 BIRTH-DEATH PROCESS 204528 Queueing Theory and Applications in Networks Assoc. Prof., Ph.D. (รศ.ดร. อน นต ผลเพ ม) Computer Engineering Department, Kasetsart University Outline 2 Birth-Death
More informationStochastic Processes
Stochastic Processes 8.445 MIT, fall 20 Mid Term Exam Solutions October 27, 20 Your Name: Alberto De Sole Exercise Max Grade Grade 5 5 2 5 5 3 5 5 4 5 5 5 5 5 6 5 5 Total 30 30 Problem :. True / False
More informationStatistics 150: Spring 2007
Statistics 150: Spring 2007 April 23, 2008 0-1 1 Limiting Probabilities If the discrete-time Markov chain with transition probabilities p ij is irreducible and positive recurrent; then the limiting probabilities
More informationBirth-Death Processes
Birth-Death Processes Birth-Death Processes: Transient Solution Poisson Process: State Distribution Poisson Process: Inter-arrival Times Dr Conor McArdle EE414 - Birth-Death Processes 1/17 Birth-Death
More informationCDA6530: Performance Models of Computers and Networks. Chapter 3: Review of Practical Stochastic Processes
CDA6530: Performance Models of Computers and Networks Chapter 3: Review of Practical Stochastic Processes Definition Stochastic process X = {X(t), t2 T} is a collection of random variables (rvs); one rv
More informationPart II: continuous time Markov chain (CTMC)
Part II: continuous time Markov chain (CTMC) Continuous time discrete state Markov process Definition (Markovian property) X(t) is a CTMC, if for any n and any sequence t 1
More informationContents LIST OF TABLES... LIST OF FIGURES... xvii. LIST OF LISTINGS... xxi PREFACE. ...xxiii
LIST OF TABLES... xv LIST OF FIGURES... xvii LIST OF LISTINGS... xxi PREFACE...xxiii CHAPTER 1. PERFORMANCE EVALUATION... 1 1.1. Performance evaluation... 1 1.2. Performance versus resources provisioning...
More informationQueueing. Chapter Continuous Time Markov Chains 2 CHAPTER 5. QUEUEING
2 CHAPTER 5. QUEUEING Chapter 5 Queueing Systems are often modeled by automata, and discrete events are transitions from one state to another. In this chapter we want to analyze such discrete events systems.
More informationAnalysis of an Infinite-Server Queue with Markovian Arrival Streams
Analysis of an Infinite-Server Queue with Markovian Arrival Streams Guidance Professor Associate Professor Assistant Professor Masao FUKUSHIMA Tetsuya TAKINE Nobuo YAMASHITA Hiroyuki MASUYAMA 1999 Graduate
More informationAssignment 3 with Reference Solutions
Assignment 3 with Reference Solutions Exercise 3.: Poisson Process Given are k independent sources s i of jobs as shown in the figure below. The interarrival time between jobs for each source is exponentially
More informationLecture 20: Reversible Processes and Queues
Lecture 20: Reversible Processes and Queues 1 Examples of reversible processes 11 Birth-death processes We define two non-negative sequences birth and death rates denoted by {λ n : n N 0 } and {µ n : n
More informationLet (Ω, F) be a measureable space. A filtration in discrete time is a sequence of. F s F t
2.2 Filtrations Let (Ω, F) be a measureable space. A filtration in discrete time is a sequence of σ algebras {F t } such that F t F and F t F t+1 for all t = 0, 1,.... In continuous time, the second condition
More informationExercises Stochastic Performance Modelling. Hamilton Institute, Summer 2010
Exercises Stochastic Performance Modelling Hamilton Institute, Summer Instruction Exercise Let X be a non-negative random variable with E[X ]
More informationLectures on Probability and Statistical Models
Lectures on Probability and Statistical Models Phil Pollett Professor of Mathematics The University of Queensland c These materials can be used for any educational purpose provided they are are not altered
More informationTheory and Applications of Stochastic Systems Lecture Exponential Martingale for Random Walk
Instructor: Victor F. Araman December 4, 2003 Theory and Applications of Stochastic Systems Lecture 0 B60.432.0 Exponential Martingale for Random Walk Let (S n : n 0) be a random walk with i.i.d. increments
More informationRandom variables. DS GA 1002 Probability and Statistics for Data Science.
Random variables DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Motivation Random variables model numerical quantities
More informationContinuous time Markov chains
Continuous time Markov chains Alejandro Ribeiro Dept. of Electrical and Systems Engineering University of Pennsylvania aribeiro@seas.upenn.edu http://www.seas.upenn.edu/users/~aribeiro/ October 16, 2017
More information8. Statistical Equilibrium and Classification of States: Discrete Time Markov Chains
8. Statistical Equilibrium and Classification of States: Discrete Time Markov Chains 8.1 Review 8.2 Statistical Equilibrium 8.3 Two-State Markov Chain 8.4 Existence of P ( ) 8.5 Classification of States
More informationTCOM 501: Networking Theory & Fundamentals. Lecture 6 February 19, 2003 Prof. Yannis A. Korilis
TCOM 50: Networking Theory & Fundamentals Lecture 6 February 9, 003 Prof. Yannis A. Korilis 6- Topics Time-Reversal of Markov Chains Reversibility Truncating a Reversible Markov Chain Burke s Theorem Queues
More informationGeometric ρ-mixing property of the interarrival times of a stationary Markovian Arrival Process
Author manuscript, published in "Journal of Applied Probability 50, 2 (2013) 598-601" Geometric ρ-mixing property of the interarrival times of a stationary Markovian Arrival Process L. Hervé and J. Ledoux
More informationHomework 4 due on Thursday, December 15 at 5 PM (hard deadline).
Large-Time Behavior for Continuous-Time Markov Chains Friday, December 02, 2011 10:58 AM Homework 4 due on Thursday, December 15 at 5 PM (hard deadline). How are formulas for large-time behavior of discrete-time
More informationN.G.Bean, D.A.Green and P.G.Taylor. University of Adelaide. Adelaide. Abstract. process of an MMPP/M/1 queue is not a MAP unless the queue is a
WHEN IS A MAP POISSON N.G.Bean, D.A.Green and P.G.Taylor Department of Applied Mathematics University of Adelaide Adelaide 55 Abstract In a recent paper, Olivier and Walrand (994) claimed that the departure
More informationIntro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks. Queuing Networks. Florence Perronnin
Queuing Networks Florence Perronnin Polytech Grenoble - UGA March 23, 27 F. Perronnin (UGA) Queuing Networks March 23, 27 / 46 Outline Introduction to Queuing Networks 2 Refresher: M/M/ queue 3 Reversibility
More informationLecture 7: Simulation of Markov Processes. Pasi Lassila Department of Communications and Networking
Lecture 7: Simulation of Markov Processes Pasi Lassila Department of Communications and Networking Contents Markov processes theory recap Elementary queuing models for data networks Simulation of Markov
More informationReview of Mathematical Concepts. Hongwei Zhang
Review of Mathematical Concepts Hongwei Zhang http://www.cs.wayne.edu/~hzhang Outline Limits of real number sequences A fixed-point theorem Probability and random processes Probability model Random variable
More informationIEOR 6711: Stochastic Models I, Fall 2003, Professor Whitt. Solutions to Final Exam: Thursday, December 18.
IEOR 6711: Stochastic Models I, Fall 23, Professor Whitt Solutions to Final Exam: Thursday, December 18. Below are six questions with several parts. Do as much as you can. Show your work. 1. Two-Pump Gas
More informationLecture 21. David Aldous. 16 October David Aldous Lecture 21
Lecture 21 David Aldous 16 October 2015 In continuous time 0 t < we specify transition rates or informally P(X (t+δ)=j X (t)=i, past ) q ij = lim δ 0 δ P(X (t + dt) = j X (t) = i) = q ij dt but note these
More informationP 1.5 X 4.5 / X 2 and (iii) The smallest value of n for
DHANALAKSHMI COLLEGE OF ENEINEERING, CHENNAI DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING MA645 PROBABILITY AND RANDOM PROCESS UNIT I : RANDOM VARIABLES PART B (6 MARKS). A random variable X
More informationThe exponential distribution and the Poisson process
The exponential distribution and the Poisson process 1-1 Exponential Distribution: Basic Facts PDF f(t) = { λe λt, t 0 0, t < 0 CDF Pr{T t) = 0 t λe λu du = 1 e λt (t 0) Mean E[T] = 1 λ Variance Var[T]
More informationAn M/M/1 Queue in Random Environment with Disasters
An M/M/1 Queue in Random Environment with Disasters Noam Paz 1 and Uri Yechiali 1,2 1 Department of Statistics and Operations Research School of Mathematical Sciences Tel Aviv University, Tel Aviv 69978,
More informationINDEX. production, see Applications, manufacturing
INDEX Absorbing barriers, 103 Ample service, see Service, ample Analyticity, of generating functions, 100, 127 Anderson Darling (AD) test, 411 Aperiodic state, 37 Applications, 2, 3 aircraft, 3 airline
More information57:022 Principles of Design II Final Exam Solutions - Spring 1997
57:022 Principles of Design II Final Exam Solutions - Spring 1997 Part: I II III IV V VI Total Possible Pts: 52 10 12 16 13 12 115 PART ONE Indicate "+" if True and "o" if False: + a. If a component's
More informationEleventh Problem Assignment
EECS April, 27 PROBLEM (2 points) The outcomes of successive flips of a particular coin are dependent and are found to be described fully by the conditional probabilities P(H n+ H n ) = P(T n+ T n ) =
More informationVariance reduction techniques
Variance reduction techniques Lecturer: Dmitri A. Moltchanov E-mail: moltchan@cs.tut.fi http://www.cs.tut.fi/kurssit/elt-53606/ OUTLINE: Simulation with a given accuracy; Variance reduction techniques;
More informationMarkov processes and queueing networks
Inria September 22, 2015 Outline Poisson processes Markov jump processes Some queueing networks The Poisson distribution (Siméon-Denis Poisson, 1781-1840) { } e λ λ n n! As prevalent as Gaussian distribution
More informationDisjointness and Additivity
Midterm 2: Format Midterm 2 Review CS70 Summer 2016 - Lecture 6D David Dinh 28 July 2016 UC Berkeley 8 questions, 190 points, 110 minutes (same as MT1). Two pages (one double-sided sheet) of handwritten
More informationMidterm 2 Review. CS70 Summer Lecture 6D. David Dinh 28 July UC Berkeley
Midterm 2 Review CS70 Summer 2016 - Lecture 6D David Dinh 28 July 2016 UC Berkeley Midterm 2: Format 8 questions, 190 points, 110 minutes (same as MT1). Two pages (one double-sided sheet) of handwritten
More informationSTAT STOCHASTIC PROCESSES. Contents
STAT 3911 - STOCHASTIC PROCESSES ANDREW TULLOCH Contents 1. Stochastic Processes 2 2. Classification of states 2 3. Limit theorems for Markov chains 4 4. First step analysis 5 5. Branching processes 5
More informationAnalysis of a tandem queueing model with GI service time at the first queue
nalysis of a tandem queueing model with GI service time at the first queue BSTRCT Zsolt Saffer Department of Telecommunications Budapest University of Technology and Economics, Budapest, Hungary safferzs@hitbmehu
More informationProbability Distributions
Lecture : Background in Probability Theory Probability Distributions The probability mass function (pmf) or probability density functions (pdf), mean, µ, variance, σ 2, and moment generating function (mgf)
More informationSome Background Information on Long-Range Dependence and Self-Similarity On the Variability of Internet Traffic Outline Introduction and Motivation Ch
On the Variability of Internet Traffic Georgios Y Lazarou Information and Telecommunication Technology Center Department of Electrical Engineering and Computer Science The University of Kansas, Lawrence
More informationELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Definition of stochastic process (random
More informationContents Preface The Exponential Distribution and the Poisson Process Introduction to Renewal Theory
Contents Preface... v 1 The Exponential Distribution and the Poisson Process... 1 1.1 Introduction... 1 1.2 The Density, the Distribution, the Tail, and the Hazard Functions... 2 1.2.1 The Hazard Function
More informationLIMITING PROBABILITY TRANSITION MATRIX OF A CONDENSED FIBONACCI TREE
International Journal of Applied Mathematics Volume 31 No. 18, 41-49 ISSN: 1311-178 (printed version); ISSN: 1314-86 (on-line version) doi: http://dx.doi.org/1.173/ijam.v31i.6 LIMITING PROBABILITY TRANSITION
More informationStochastic Simulation
Stochastic Simulation Jan-Pieter Dorsman 1 & Michel Mandjes 1,2,3 1 Korteweg-de Vries Institute for Mathematics, University of Amsterdam 2 CWI, Amsterdam 3 Eurandom, Eindhoven University of Amsterdam,
More informationStat 516, Homework 1
Stat 516, Homework 1 Due date: October 7 1. Consider an urn with n distinct balls numbered 1,..., n. We sample balls from the urn with replacement. Let N be the number of draws until we encounter a ball
More informationQueueing Review. Christos Alexopoulos and Dave Goldsman 10/6/16. (mostly from BCNN) Georgia Institute of Technology, Atlanta, GA, USA
1 / 24 Queueing Review (mostly from BCNN) Christos Alexopoulos and Dave Goldsman Georgia Institute of Technology, Atlanta, GA, USA 10/6/16 2 / 24 Outline 1 Introduction 2 Queueing Notation 3 Transient
More informationStochastic Processes. Theory for Applications. Robert G. Gallager CAMBRIDGE UNIVERSITY PRESS
Stochastic Processes Theory for Applications Robert G. Gallager CAMBRIDGE UNIVERSITY PRESS Contents Preface page xv Swgg&sfzoMj ybr zmjfr%cforj owf fmdy xix Acknowledgements xxi 1 Introduction and review
More informationTime Reversibility and Burke s Theorem
Queuing Analysis: Time Reversibility and Burke s Theorem Hongwei Zhang http://www.cs.wayne.edu/~hzhang Acknowledgement: this lecture is partially based on the slides of Dr. Yannis A. Korilis. Outline Time-Reversal
More informationJordan normal form notes (version date: 11/21/07)
Jordan normal form notes (version date: /2/7) If A has an eigenbasis {u,, u n }, ie a basis made up of eigenvectors, so that Au j = λ j u j, then A is diagonal with respect to that basis To see this, let
More informationVariance reduction techniques
Variance reduction techniques Lecturer: Dmitri A. Moltchanov E-mail: moltchan@cs.tut.fi http://www.cs.tut.fi/ moltchan/modsim/ http://www.cs.tut.fi/kurssit/tlt-2706/ OUTLINE: Simulation with a given confidence;
More informationQueueing Theory I Summary! Little s Law! Queueing System Notation! Stationary Analysis of Elementary Queueing Systems " M/M/1 " M/M/m " M/M/1/K "
Queueing Theory I Summary Little s Law Queueing System Notation Stationary Analysis of Elementary Queueing Systems " M/M/1 " M/M/m " M/M/1/K " Little s Law a(t): the process that counts the number of arrivals
More informationContinuous-time Markov Chains
Continuous-time Markov Chains Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ October 23, 2017
More informationISyE 6650 Test 2 Solutions
1 NAME ISyE 665 Test 2 Solutions Summer 2 This test is open notes, open books. The test is two hours long. 1. Consider an M/M/3/4 queueing system in steady state with arrival rate λ = 3 and individual
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions 3.6 Conditional
More informationDISCRETE STOCHASTIC PROCESSES Draft of 2nd Edition
DISCRETE STOCHASTIC PROCESSES Draft of 2nd Edition R. G. Gallager January 31, 2011 i ii Preface These notes are a draft of a major rewrite of a text [9] of the same name. The notes and the text are outgrowths
More informationBirth and Death Processes. Birth and Death Processes. Linear Growth with Immigration. Limiting Behaviour for Birth and Death Processes
DTU Informatics 247 Stochastic Processes 6, October 27 Today: Limiting behaviour of birth and death processes Birth and death processes with absorbing states Finite state continuous time Markov chains
More informationMarkov Chain Monte Carlo The Metropolis-Hastings Algorithm
Markov Chain Monte Carlo The Metropolis-Hastings Algorithm Anthony Trubiano April 11th, 2018 1 Introduction Markov Chain Monte Carlo (MCMC) methods are a class of algorithms for sampling from a probability
More information6 Markov Chain Monte Carlo (MCMC)
6 Markov Chain Monte Carlo (MCMC) The underlying idea in MCMC is to replace the iid samples of basic MC methods, with dependent samples from an ergodic Markov chain, whose limiting (stationary) distribution
More informationMatrix analytic methods. Lecture 1: Structured Markov chains and their stationary distribution
1/29 Matrix analytic methods Lecture 1: Structured Markov chains and their stationary distribution Sophie Hautphenne and David Stanford (with thanks to Guy Latouche, U. Brussels and Peter Taylor, U. Melbourne
More information18.440: Lecture 33 Markov Chains
18.440: Lecture 33 Markov Chains Scott Sheffield MIT 1 Outline Markov chains Examples Ergodicity and stationarity 2 Outline Markov chains Examples Ergodicity and stationarity 3 Markov chains Consider a
More informationBIRTH DEATH PROCESSES AND QUEUEING SYSTEMS
BIRTH DEATH PROCESSES AND QUEUEING SYSTEMS Andrea Bobbio Anno Accademico 999-2000 Queueing Systems 2 Notation for Queueing Systems /λ mean time between arrivals S = /µ ρ = λ/µ N mean service time traffic
More informationNumerical Transform Inversion to Analyze Teletraffic Models
Numerical Transform Inversion to Analyze Teletraffic Models Gagan L. Choudhury, a David M. Lucantoni a and Ward Whitt b a AT&T Bell Laboratories, Holmdel, NJ 07733-3030, USA b AT&T Bell Laboratories, Murray
More informationIE 303 Discrete-Event Simulation
IE 303 Discrete-Event Simulation 1 L E C T U R E 5 : P R O B A B I L I T Y R E V I E W Review of the Last Lecture Random Variables Probability Density (Mass) Functions Cumulative Density Function Discrete
More informationGlossary availability cellular manufacturing closed queueing network coefficient of variation (CV) conditional probability CONWIP
Glossary availability The long-run average fraction of time that the processor is available for processing jobs, denoted by a (p. 113). cellular manufacturing The concept of organizing the factory into
More informationMarkov Chains and MCMC
Markov Chains and MCMC Markov chains Let S = {1, 2,..., N} be a finite set consisting of N states. A Markov chain Y 0, Y 1, Y 2,... is a sequence of random variables, with Y t S for all points in time
More informationJ. MEDHI STOCHASTIC MODELS IN QUEUEING THEORY
J. MEDHI STOCHASTIC MODELS IN QUEUEING THEORY SECOND EDITION ACADEMIC PRESS An imprint of Elsevier Science Amsterdam Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo Contents
More informationCS 237: Probability in Computing
CS 237: Probability in Computing Wayne Snyder Computer Science Department Boston University Lecture 13: Normal Distribution Exponential Distribution Recall that the Normal Distribution is given by an explicit
More informationTHE QUEEN S UNIVERSITY OF BELFAST
THE QUEEN S UNIVERSITY OF BELFAST 0SOR20 Level 2 Examination Statistics and Operational Research 20 Probability and Distribution Theory Wednesday 4 August 2002 2.30 pm 5.30 pm Examiners { Professor R M
More informationContinuous Time Processes
page 102 Chapter 7 Continuous Time Processes 7.1 Introduction In a continuous time stochastic process (with discrete state space), a change of state can occur at any time instant. The associated point
More informationQueueing Review. Christos Alexopoulos and Dave Goldsman 10/25/17. (mostly from BCNN) Georgia Institute of Technology, Atlanta, GA, USA
1 / 26 Queueing Review (mostly from BCNN) Christos Alexopoulos and Dave Goldsman Georgia Institute of Technology, Atlanta, GA, USA 10/25/17 2 / 26 Outline 1 Introduction 2 Queueing Notation 3 Transient
More informationStochastic modelling of epidemic spread
Stochastic modelling of epidemic spread Julien Arino Centre for Research on Inner City Health St Michael s Hospital Toronto On leave from Department of Mathematics University of Manitoba Julien Arino@umanitoba.ca
More information6 Continuous-Time Birth and Death Chains
6 Continuous-Time Birth and Death Chains Angela Peace Biomathematics II MATH 5355 Spring 2017 Lecture notes follow: Allen, Linda JS. An introduction to stochastic processes with applications to biology.
More informationLink Models for Circuit Switching
Link Models for Circuit Switching The basis of traffic engineering for telecommunication networks is the Erlang loss function. It basically allows us to determine the amount of telephone traffic that can
More informationHEAVY-TRAFFIC ASYMPTOTIC EXPANSIONS FOR THE ASYMPTOTIC DECAY RATES IN THE BMAP/G/1 QUEUE
HEAVY-TRAFFIC ASYMPTOTIC EXPANSIONS FOR THE ASYMPTOTIC DECAY RATES IN THE BMAP/G/1 QUEUE Gagan L. CHOUDHURY AT&T Bell Laboratories Holmdel, NJ 07733-3030 Ward WHITT AT&T Bell Laboratories Murray Hill,
More informationChapter 6: Random Processes 1
Chapter 6: Random Processes 1 Yunghsiang S. Han Graduate Institute of Communication Engineering, National Taipei University Taiwan E-mail: yshan@mail.ntpu.edu.tw 1 Modified from the lecture notes by Prof.
More informationSYMBOLS AND ABBREVIATIONS
APPENDIX A SYMBOLS AND ABBREVIATIONS This appendix contains definitions of common symbols and abbreviations used frequently and consistently throughout the text. Symbols that are used only occasionally
More information1.225J J (ESD 205) Transportation Flow Systems
1.225J J (ESD 25) Transportation Flow Systems Lecture 9 Simulation Models Prof. Ismail Chabini and Prof. Amedeo R. Odoni Lecture 9 Outline About this lecture: It is based on R16. Only material covered
More information