Approximating Deterministic Changes to Ph(t)/Ph(t)/1/c and Ph(t)/M(t)/s/c Queueing Models

Transcription

1 Approximating Deterministic Changes to Ph(t)/Ph(t)/1/c and Ph(t)/M(t)/s/c Queueing Models Aditya Umesh Kulkarni Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Master of Science in Industrial and Systems Engineering Michael R. Taaffe, Chair Raghu Pasupathy Ebru K. Bish May 25, 212 Blacksburg, Virginia Keywords: Queueing, phase-type, time-varying queues, Polya Eggenberger Copyright 212, Aditya Umesh Kulkarni

2 Approximating Deterministic Changes to Ph(t)/Ph(t)/1/c and Ph(t)/M(t)/s/c Queueing Models Aditya Umesh Kulkarni (ABSTRACT) A deterministic change to a time-varying queueing model is described as either changing the number of entities, the queue capacity, or the number of servers in the system at selected times. We use a surrogate distribution for N(t), the number of entities in the system at time t, to approximate deterministic changes to the Ph(t)/Ph(t)/1/c and the Ph(t)/M(t)/s/c queueing models. We develop a solution technique to minimize the number of state probabilities to be approximated.

3 Dedication This work is dedicated to my fiancé, Ashima Athri, without whose constant encouragement and motivation it would not have been possible, and to my parents, Umesh Kulkarni and Tara Umesh, whose sole purpose in life has been my education. iii

4 Acknowledgments I would like to thank Dr. Michael R. Taaffe for his constant support, encouragement, his ability to put his faith in me, and for gracefully controlling my youthful audacity. Special thanks to my committee members, Dr. Raghu Pasupathy and Dr. Ebru K. Bish, who supported me throughout my thesis work. iv

5 Contents 1 Introduction Motivation M(t)/M(t)/1/c example Polya Eggenberger distribution Deterministic addition to N(t) Deterministic deletion from N(t) Deterministic increases in queue capacity c Deterministic decrease in queue capacity c Updating E[N(t), N(t) β] Algorithm SSMQ Conclusion v

6 2 Deterministic changes to the Ph(t)/Ph(t)/1/c queueing model Deterministic addition to N(t) Deterministic deletion from N(t) Deterministic increase in queue capacity c Deterministic decrease in queue capacity c Updating idle state KFEs and PMDEs Algorithm SSFC Conclusion Deterministic changes to the Ph(t)/M(t)/s/c queueing model Deterministic addition to N(t) Deterministic deletion from N(t) Deterministic increase in queue capacity c s Deterministic decrease in queue capacity c s Deterministic addition to s Deterministic deletion from s Updating required values Algorithm MSFC vi

7 3.9 Conclusion Tests and Results Ph(t)/Ph(t)/1/c queueing model results Ph(t)/M(t)/s/c queueing model results Conclusion 85 Bibliography 85 Appix I 88 Appix II 18 Appix III 259 vii

8 List of Figures 1.1 Approximating E[N(t)], Ph(t)/Ph(t)/1/c queueing model (Test Case 5) Approximating StdDev[N(t)], Ph(t)/Ph(t)/1/c queueing model (Test Case 5) Approximating E[N(t)], Ph(t)/Ph(t)/1/c queueing model (Test Case 15) Approximating StdDev[N(t)], Ph(t)/Ph(t)/1/c queueing model (Test Case 15) Approximating E[N(t)], Ph(t)/M(t)/s/c queueing model (Test Case 5) Approximating StdDev[N(t)], Ph(t)/M(t)/s/c queueing model (Test Case 5) Approximating E[N(t)], Ph(t)/M(t)/s/c queueing model (Test Case 15) Approximating StdDev[N(t)], Ph(t)/M(t)/s/c queueing model (Test Case 15) Simple decision tree for deterministic changes to M(t)/M(t)/1/c queueing model Decision tree for deterministic changes to Ph(t)/Ph(t)/1/c queueing model Decision tree for Ω 1,i, for the Ph(t)/M(t)/s/c queueing model viii

9 3.2 Decision tree for Ω 2,i, for the Ph(t)/M(t)/s/c queueing model E[N(t)] vs. t and errors for Case StdDev[N(t)] vs. t and errors for Case E[N(t)] vs. t and errors for Case SD[N(t)] vs. t and errors for Case E[N(t)] vs. t and errors SD[N(t)] vs. t and errors E[N(t)] vs. t and errors SD[N(t)] vs. t and errors ix

10 List of Tables 1.1 Number of state probabilities approximated using Method Ph(t)/Ph(t)/1/c queueing model information E[N(t)] related errors for the Ph(t)/Ph(t)/1/c queueing model StdDev[N(t)] related errors for the Ph(t)/Ph(t)/1/c queueing model Ph(t)/M(t)/s/c queueing model information E[N(t)] related errors for the Ph(t)/M(t)/s/c queueing model StdDev[N(t)] related errors for the Ph(t)/M(t)/s/c queueing model x

11 Chapter 1 Introduction The state probabilities for a time-varying Phase-type queueing model can be obtained by numerically integrating the full set of Kolmogorov forward equations (KFEs) for the queueing model. Quantitative measures, such as the expectation and variance of the number of entities in the system, are computed from the state probabilities obtained via numerical integration of KFEs. Deterministic change to a time-varying Phase-type queueing model is defined as deterministically changing the number of entities in the system, or the number of servers, or the queue capacity, at selected times. Deterministic changes made to the system at selected times causes an instantaneous change in the state probabilities, at these times. This requires the KFEs to be updated at the times of deterministic changes to the system. Mapping the state of a time-varying Phase-type queueing model, that undergoes deterministic changes, with respect to time, by using the KFEs, requires computational effort. The computational effort increases as the number of KFEs increase. We ext the work of Taaffe and Ong 1

12 2 [1, 8] and approximate the p th moment of N(t), where N(t) is a random variable that denotes the number of entities in the system at time t, when deterministic changes are made to Ph(t)/Ph(t)/1/c and Ph(t)/M(t)/s/c queueing models. 1.1 Motivation Taaffe and Ong [1, 8] showed, that the p th moment of N(t) is well approximated using a state-space partitioning and surrogate distribution approximation (SDA) approach for the Ph(t)/Ph(t)/1/c and Ph(t)/M(t)/s/c queueing model, where the time between arrivals has a continuous positive valued distribution, the service time has a continuous positive valued distribution, the number of servers for each model is constant in the time interval of interest, and the queue capacity for each model is constant in the time interval of interest. Such a scenario is an extreme case. The general case is where the time between arrivals and service time have indepent continuous positive valued distribution, and deterministic changes are made to the number of entities in the system, the number of servers in the system, or the queue capacity of the system, at selected times. We developed software to approximate p th moment of N(t) for the general case. Figure 1.1 and Figure 1.2 show the approximation and associated absolute error for E[N(t)] and StdDev[N(t)], respectively, for the Ph(t)/Ph(t)/1/c queueing model, where the time between arrivals and service time are equal to zero and deterministic changes are made to the system at selected times in the time interval of interest. Figure 1.3 and Figure 1.4 show the approximation and associated

13 3 absolute error for E[N(t)] and StdDev[N(t)], respectively, for the Ph(t)/Ph(t)/1/c queueing model, where the time between arrivals and service time have indepent time-varying Phase-type distribution, and deterministic changes are made to the system at selected times in the time interval of interest. In the figures, the solid line denotes the actual value vs. t and the dotted line denotes the approximated value vs. t, where t lies in the time interval of interest. Absolute error is defined as: Absolute Error = Actual value - Approximated value. Figure 1.5 and Figure 1.6 show the approximation and associated absolute error for E[N(t)] and StdDev[N(t)], respectively, for the Ph(t)/M(t)/s/c queueing model, where the time between arrivals and service time are equal to zero, and deterministic changes are made to the system at selected times in the time interval of interest. Figure 1.7 and Figure 1.8 show the approximation and associated absolute error for E[N(t)] and StdDev[N(t)], respectively, for the Ph(t)/M(t)/s/c queueing model, where the time between arrivals has a time-varying Phase-type distribution, service time has a time-varying exponential distribution, and deterministic changes are made to the system at selected times in the time interval of interest. The SDA method is now known as the Closure Approximation [4]. We refer to it as the SDA method for reasons mentioned later. To develop intuition, we study deterministic changes made to the M(t)/M(t)/1/c queueing model. We present a mapping scheme for the state probabilities, when deterministic changes are made to the M(t)/M(t)/1/c queueing model, and use it to develop a solution technique.

14 4 8 Figure 1.1: Approximating E[N(t)], Ph(t)/Ph(t)/1/c queueing model (Test Case 5) E[N(t)] vs. t x 1 14 Absolute Error E[N(t)] vs. t

15 5 Figure 1.2: Approximating StdDev[N(t)], Ph(t)/Ph(t)/1/c queueing model (Test Case 5) 2.5 x 1 6 StdDev[N(t)] vs. t x 1 6 Absolute Error StdDev[N(t)] vs. t

16 Figure 1.3: Approximating E[N(t)], Ph(t)/Ph(t)/1/c queueing model (Test Case 15) E[N(t)] vs. t.5 Absolute Error E[N(t)] vs. t

17 7 Figure 1.4: Approximating StdDev[N(t)], Ph(t)/Ph(t)/1/c queueing model (Test Case 15) 8 StdDev[N(t)] vs. t Absolute Error StdDev[N(t)] vs. t

18 8 3 Figure 1.5: Approximating E[N(t)], Ph(t)/M(t)/s/c queueing model (Test Case 5) E[N(t)] vs. t x 1 15 Absolute Error E[N(t)] vs. t 3 2 1

19 9 Figure 1.6: Approximating StdDev[N(t)], Ph(t)/M(t)/s/c queueing model (Test Case 5) 5 x 1 7 StdDev[N(t)] vs. t x 1 7 Absolute Error StdDev[N(t)] vs. t

20 1 2 Figure 1.7: Approximating E[N(t)], Ph(t)/M(t)/s/c queueing model (Test Case 15) E[N(t)] vs. t Absolute Error E[N(t)] vs. t

21 11 Figure 1.8: Approximating StdDev[N(t)], Ph(t)/M(t)/s/c queueing model (Test Case 15) StdDev[N(t)] vs. t.35 Absolute Error StdDev[N(t)] vs. t M(t)/M(t)/1/c example The M(t)/M(t)/1/c queueing model is a specific case of the Ph(t)/Ph(t)/1/c queueing model, where the number of arrival phases and the number of service phases are both equal to one.

22 12 See Neuts [3], Johnson and Taaffe [1, 2], for a deeper understanding of the Phase-type distribution. Let λ(t) denote the arrival rate at time t. The service process has a timedepent exponential distribution, with rate µ(t). Let P i (t) denote P[N(t) = i], and let P i (t) denote dp[n(t) = i]/dt. Let E[N p (t)] denote the p th moment of N(t) at time t and let E [N p (t)] denote de[n p (t)]/dt. The Kolmogorov forward equations for the M(t)/M(t)/1/c queueing model are: P (t) = λ(t)p (t) + µ(t)p 1 (t) P i (t) = (λ(t) + µ(t))p i (t) + λ(t)p i 1 (t) + µ(t)p i+1 (t) for i = 1, 2,..., c 1 P c(t) = µ(t)p c (t) + λ(t)p c 1 (t). We obtain the value of E[N p (t)], where t lies in the time interval of interest, by numerically integrating the p th moment differential equation given by equation (1.1), over the time interval of interest. E [N p (t)] = c n= n p P i (t) (1.1) With simple algebra and index shifting we can simplify the right-hand side (RHS) of equation (1.1), such that the RHS of equation (1.1) contains only state probabilities and E[N q (t)], where q p. Theorem 1.1 p 1 ( ) E [N p (t)] = λ(t) p (E[N (q) (t)] c q P c (t) ) q= q p 1 ( ) +µ(t) p ( 1) p q E[N (q) (t)] ( 1) p P (t). (1.2) q q=

23 13 See Appix I for proof of equation (1.2). The SDA approach described by Taaffe and Ong [1, 8] involves using the Polya Eggenberger random variable, X t, as a surrogate random variable for N(t). We match the first two moments of N(t) to the first two moments of X t, which has a Polya Eggenberger distribution with parameters c, θ t and γ t (PE(c, θ t, γ t )). Once the parameters θ t and γ t are determined, we approximate the state probabilities P c (t) and P (t) by using P[X t = c] and P[X t = ], where X t PE(c, θ t, γ t ). The approximated probabilities are then used to close equation (1.2). Note, for the M(t)/M(t)/1/c queueing model, we numerically integrate at least the first and the second moment differential equations of N(t). The first and second moment differential equations for M(t)/M(t)/1/c queueing model are: E [N(t)] = λ(t)[1 P c (t)] µ(t)[1 P (t)] (1.3) E [N 2 (t)] = λ(t)[2e[n (1) (t) + 1 (2c + 1)P c (t)] µ(t)[2e[n (1) (t)] (1 P (t))]. (1.4) We obtain the approximation for E[N(t)] and E[N 2 (t)] by numerically integrating the moment differential equations (1.3) and (1.4), and using the approximate values of the state probabilities P (t) and P c (t) Polya Eggenberger distribution The Polya Eggenberger distribution is a special case of a Polya Urn Model. Let there be b black balls and r red balls in an urn. Define a trial as randomly choosing a ball from the urn,

24 14 noting the color of the ball, and returning the ball to the urn along with s balls of the same color. Define a success as choosing a black ball. Let X be a random variable that denotes the number of successes in n trails. The probability mass function (p.m.f) of X is P[X = i] = ( ) n i i 1 j= n i 1 (θ + jγ) j= n 1 j= (1 + jγ) (1 θ + jγ) where θ = b b + r and γ = s b + r. Thus X has a Polya Eggenberger distribution with the parameters n, γ and θ. The first two moments of X are: E[X] = cθ E[X 2 ] = cθ(c(θ + γ) + (1 θ)). (1 + γ) We match the first two moments of N(t) to the first two moments of X and evaluate θ and γ as: θ = E[N(t)] c γ = E[N(t)](E[N(t)] + (1 θ)) E[N 2 (t)]. E[N 2 (t)] ce[n(t)] P (t) is approximated by P[X = ] and P c (t) is approximated by P[X = c]. We augment this standard definition of the Polya Eggenberger distribution for the case of Var[X] = and E[X], n. If Var[X] =, then P[X = E[X]] = 1.

25 Deterministic addition to N(t) Let k (where k is a positive integer) entities be deterministically added to the system at time t ɛ, where t ɛ lies in the time interval of interest. Let the system capacity at time t ɛ be c. The system capacity, c, is finite, and we assume that deterministic additions cannot increase the number of entities in the system beyond c. Thus if k c, then k = c. Let P i (t ɛ ) denote P[N(t ɛ ) = i] and P i (t + ɛ ) denote P[N(t + ɛ ) = i]. The mapping scheme for state probabilities at time t ɛ is: P (t + ɛ ) =. P k 1 (t + ɛ ) = P k (t + ɛ ) = P (t ɛ ). P c 1 (t + ɛ ) = P c k 1 (t ɛ ) P c (t + ɛ ) = c i=c k P i (t ɛ ). The mapping scheme presented above is used to update the KFEs. This instantaneous change in the state probabilities of N(t ɛ ), leads to an instantaneous change in the moments of N(t ɛ ). The p th moment differential equation of N(t ɛ ) is updated. We update the value of E[N p (t ɛ )] as follows:

26 16 E[N p (t + ɛ )] = = = = c i p P i (t + ɛ ) i= c i p P i (t + ɛ ) i=k c i p P i k (t ɛ ) + c c p P j (t ɛ ) i=k j=c k+1 c k (i + k) p P i (t ɛ ) + c c p P j (t ɛ ). (1.5) i= j=c k+1 There are two methods to evaluate equation (1.5) without numerically integrating the KFEs to obtain the values of state probabilities at time t ɛ. For the first method we approximate the state probabilities of N(t ɛ ) with X t ɛ, where X t ɛ PE(c, θ t ɛ, γ t ɛ ). For the second method, observe that, equation (1.5) can also be written as: E[N p (t + ɛ )] = p q= ( ) p (k) p q E[N q (t ɛ ), N(t ɛ ) c k] q +c p E[N (t ɛ ), N(t ɛ ) c k + 1]. (1.6) The term E[N q (t ɛ ), N(t ɛ ) c k] is a partial-moment of N(t ɛ ), and its value is obtained by numerically integrating the partial-moment differential equations de[n q (t), N(t) c k]/dt, for q {,..., p}, over the time interval of interest. The name closure approximation suggests that we are closing differential equations by approximating state probabilities. In order to evaluate (1.5), by Method 1, we approximate the state probabilities of N(t ɛ ) with X t ɛ PE(c, θ t ɛ, γ t ɛ ). Hence we refer to the method of matching moments of N(t) to a surrogate random variable X t, Polya Eggenberger with three parameters, as the SDA method.

27 17 Method 1: Approximate state probabilities To minimize the number of approximated state probabilities, we use the values of E[N q (t ɛ )], where q {1,..., p}. E[N p (t + ɛ )] = = or E[N p (t + ɛ )] = = c (i + k) p P i (t ɛ ) i= + c [c p (j + k) p ] P j (t ɛ ) j=c k+1 ( ) p p (k) p q E[N q (t ɛ )] q= q + c [c p (j + k) p ] P j (t ɛ ) when k c k + 1 (1.7) j=c k+1 c k c [(i + k) p c p ] P i (t ɛ ) + c p P j (t ɛ ) i= j= c k i= [(i + k) p c p ] P i (t ɛ ) + c p when k > c k + 1. (1.8) Using Method 1 the maximum number of approximated state probabilities is (c + 1)/2. Method 2: Partial-Moment Differential Equations Let E [N p (t) β] denote de[n p (t) β]/dt, where < β < c. Theorem 1.2 E [N p (t) β] = δ β, λ(t) ( δ p, ( P (t)) + δ p, ) p 1 ( ) +δ β, δ β,c λ(t) p E[N q (t), N(t) β] (β + 1) p P β (t) q= q p 1 ( ) p p 1 +δ β,c λ(t) E[N (q) (t)] c q P c (t) q= q q= +δ β, µ(t) ( δ p, P 1 (t) + δ p, )

28 p 1 +δ β, µ(t) q= ( ) p ( 1) p q E[N q (t), N(t) β] q +δ β, µ(t) ( δ β,c β p P β+1 (t) δ p, ( 1) p P (t) ) (1.9) 18 where δ a,b = 1 if a = b if a b δ a,b = 1 δ a,b < β < c. Refer to Appix I for proof of Theorem 1.2. Numerical integration of equation (1.9) requires the values of P (t), P c (t), P β (t) and P β+1 (t). Note, by using the SDA approach we approximate P (t) and P c (t) in order to numerically integrate the first and second moment differential equations of N(t). To approximate P β (t), define a random variable W t such that W t PE(β, θ t1, γ t1 ). Match the conditional moments E[N(t) N(t) β] and E[N 2 (t) N(t) β] to the first and second moment of W t, respectively, and evaluate θ t1 and γ t1. The approximate value of P β (t) is given by P[W t = β]. To approximate P β+1 (t), define a random variable Y t such that Y t PE(c β 1, θ t2, γ t2 ). Match E[N(t) N(t) > β] (β + 1) and E[N 2 (t) N(t) > β] 2(β + 1)(E[N(t) N(t) > β] (β + 1)) (β + 1) 2 to the first and second moment of Y t, respectively. The approximate value of P β+1 (t) is given by P[Y t = ].

29 Deterministic deletion from N(t) Let k (where k is a positive integer) entities be deterministically deleted from the system at time t ɛ, where t ɛ lies in the time interval of interest. Let the system capacity at time t ɛ be c. There can be at most c entities in the system at time t ɛ. If k c, then k = c. Let P i (t ɛ ) denote P[N(t ɛ ) = i] and P i (t + ɛ ) denote P[N(t + ɛ ) = i]. The mapping scheme for state probabilities at time t ɛ is: P c (t + ɛ ) =. P c k (t + ɛ ) = P c (t ɛ ). P 1 (t + ɛ ) = P k+1 (t ɛ ) P (t + ɛ ) = k P i (t ɛ ). i= At time t + ɛ the p th moment of N(t + ɛ ) is E[N p (t + ɛ )] = = c i p P i (t + ɛ ) i= c k i p P i (t + ɛ ) i= k = δ p, P i (t ɛ ) + i p P i+k (t ɛ ) i= k 1 = δ p, i= k 1 = δ p, i= P i (t ɛ ) + P i (t ɛ ) + c k i=1 c k i= c i=k i p P i+k (t ɛ ) (i k) p P i (t ɛ ). (1.1)

30 2 Equation (1.1), can be approximated by either Method 1 or Method 2. Method 1 k 1 E[N p (t + c ɛ )] = δ p, P i (t ɛ ) + (i k) p P i (t ɛ ) when k > c k + 1 (1.11) i= i=k or k 1 E[N p (t + ɛ )] = δ p, i= k 1 i= k 1 = δ p, i= k 1 i= c P i (t ɛ ) + (i k) p P i+k (t ɛ ) i= (i k) p P i (t ɛ ) P i (t ɛ ) + p q= ( ) p ( k) p q E[N q (t ɛ )] q (i k) p P i (t ɛ ) when k c k + 1. (1.12) Method 2 k 1 E[N p (t + c ɛ )] = δ p, P i (t ɛ ) + (i k) p P i (t ɛ ) i= i=k k 1 c = δ p, P i (t ɛ ) + (i k) p P i (t ɛ ) i= i= k 1 (i k) p P i (t ɛ ) i= p = δ p, E[N (t ɛ ), N(t ɛ ) k 1] + q= p q= ( ) p ( k) p q E[N q (t ɛ )] q ( ) p ( k) p q E[N q (t ɛ ), N(t ɛ ) k 1] (1.13) q Next consider deterministic changes made to the queue capacity c 1.

31 Deterministic increases in queue capacity c 1 Let the system capacity at time t ɛ be c. Let the queue capacity, c 1, be increased deterministically by k (where k is a positive integer) at time t ɛ. Let P i (t ɛ ) denote P[N(t ɛ ) = i] and P i (t + ɛ ) denote P[N(t + ɛ ) = i]. The mapping scheme for state probabilities at time t ɛ is: P (t + ɛ ) = P (t ɛ ). P c (t + ɛ ) = P c (t ɛ ) P c+1 (t + ɛ ) =. P c+k (t + ɛ ) =. At time t + ɛ the p th moment of N(t + ɛ ) is given by: E[N p (t + ɛ )] = = c+k i= c i= i p P i (t + ɛ ) i p P i (t ɛ ) = E[N p (t ɛ )]. (1.14) A deterministic increase in queue capacity c 1 creates new states for the system and the state probability values, for these new states, are all equal to zero. Deterministic increase in queue capacity does not change the probability values of the existing states.

32 Deterministic decrease in queue capacity c 1 Let the system capacity at time t ɛ be c. Let the queue capacity c 1, where c 1, be decreased deterministically by k (where k is a positive integer) at time t ɛ. The queue capacity c 1 is finite. If k > c 1, then k = c 1. Let P i (t ɛ ) denote P[N(t ɛ ) = i] and P i (t + ɛ ) denote P[N(t + ɛ ) = i]. The mapping scheme for state probabilities at time t ɛ is: P c (t + ɛ ) =. P c k+1 (t + ɛ ) = P c k (t + ɛ ) = P c k (t ɛ ) + c P i (t ɛ ) P c k 1 (t + ɛ ) = P c k 1 (t ɛ ). P (t + ɛ ) = P (t ɛ ). i=c k+1 At time t + ɛ the p th moment of N(t + ɛ ) is: E[N p (t + ɛ )] = = = c i p P i (t + ɛ ) i= c k 1 i p P i (t ɛ ) + (c k) p c P i (t ɛ ) i= i=c k c k i p P i (t ɛ ) + (c k) p c P i (t ɛ ). (1.15) i= i=c k+1 Method 1 or Method 2 can be used to evaluate equation (1.15).

33 23 Method 1 E[N p (t + ɛ )] = c k c i p P i (t ɛ ) + (c k) p P i (t ɛ ) i= i=c k+1 = c i p P i (t ɛ ) + c ((c k) p i p ) P i (t ɛ ) i= i=c k+1 = E[N p (t ɛ )] or E[N p (t + ɛ )] = = c + ((c k) p i p ) P i (t ɛ ) when k c k + 1 (1.16) c k i=c k+1 c (i p (c k) p ) P i (t ɛ ) + (c k) p P i (t ɛ ) i= i= c k i= (i p (c k) p ) P i (t ɛ ) + (c k) p when k > c k + 1. (1.17) Method 2 c k E[N p (t + c ɛ )] = i p P i (t ɛ ) + (c k) p P i (t ɛ ) i= i=c k+1 = E[N p (t ɛ ), N(t ɛ ) c k] +(c k) ( p 1 P[N(t ɛ ) c k]. ) (1.18) Method 2 requires the value of the partial moment E[N p (t), N(t) β] to update E[N p (t)]. The partial moment E[N p (t), N(t) β] deps on β, which in turn deps on the type of deterministic changes to be made. Since β is not fixed for every case, the states β and β + 1, whose probabilities are approximated, are not fixed. We numerically integrate one differential equation for each unique value of β. We now consider how to suitably update E[N(t), N(t) β].

34 Updating E[N(t), N(t) β] Deterministic increases in queue capacity c 1 does not require E[N p (t), N(t) β] to be updated since the probabilities of the existing states do not change (the probabilities of the new states created are equal to zero). Excluding deterministic increases in queue capacity, we have three possible types of deterministic changes, where E[N p (t), N(t) β] needs to be updated. They are deterministic additions to N(t), deterministic deletions from N(t) and deterministic decreases in c 1. We refer to these as Addition, Deletion and Decrease respectively. Let the time interval of interest be (t, t f ). Assume an Addition occurs at times t 1, t 2 and t 3 of magnitude k 1, k 2 and k 3, where t 1, t 2, t 3 < t f and k 1 k 2 k 3. Consider the last deterministic change first and observe that E[N p (t 3 ), N(t 3 ) c k 3 ] is required to update E[N p (t 3 )]. Similarly, at time t 2 and t 1, E[N p (t 2 ), N(t 2 ) c k 2 ] and E[N p (t 1 ), N(t 1 ) c k 1 ] are required to update E[N p (t)] at times t 2 and t 1, respectively. Thus, in this case we numerically integrate de[n p (t), N(t) c k 1 ]/dt from time to t 1, de[n p (t), N(t) c k 2 ]/dt from time to t 2, and de[n p (t), N(t) c k 3 ]/dt from time to t 3. The first deterministic change occurs at time t 1, thus we do not update E[N p (t), N(t) c k 1 ], unlike E[N p (t), N(t) c k 2 ] and E[N p (t), N(t) c k 3 ], which do need to be updated. We update E[N p (t), N(t) c k 3 ] at times t 1 and t 2. Similarly, we update E[N p (t), N(t) c k 2 ] at time t 1. At time t 2, the mapping scheme for state

35 25 probabilities is: P (t + 2 ) =. P k2 1(t + 2 ) = P k2 (t + 2 ) = P (t 2 ). P c 1 (t + 2 ) = P c k2 1(t 2 ) P c (t + 2 ) = c i=c k 2 P i (t 2 ). thus E[N p (t 2 + ), N(t 2 + ) c k 3 ] = c k 3 i= i p P i (t 2 + ) = if c k 3 k 2 1 or = = = = c k 3 i=k 2 i p P i (t 2 c k 3 + ) i=k 2 i p P i k2 (t 2 c k 3 k 2 i= p q= ( p q ) (i + k 2 ) p P i (t 2 ) (1.19) ) E[N q (t 2 ), N(t 2 ) c k 3 k 2 ] if c k 3 k 2 (1.2) Observe that E[N p (t 2 + ), N(t 2 + ) c k 3 ] can either be evaluated by numerically integrating de[n q (t), N(t) c k 3 k 2 ]/dt, where q {,..., p}, or by approximating state proba-

36 26 bilities. Numerically integrating de[n q (t), N(t) c k 3 k 2 ]/dt, when c k 3 k 3 k 1 requires E[N q (t 1 ), N(t 1 ) c k 3 k 2 k 1 ] in order to update E[N q (t), N(t) c k 3 k 2 ] at time t 1. Observe that, in this example, the total number of differential equations numerically integrated deps on the number of deterministic changes made, and the type of deterministic changes made in the time interval of interest. We refer to one type of deterministic change followed by the same or another type of deterministic change as a change-pair. There are 16 change-pairs for the M(t)/M(t)/1/c model. Of these only 9 change-pairs require E[N p (t)] to be updated at the times of both deterministic changes in the change-pair. For a large number of deterministic changes in the time interval of interest, the numerical integration of a large number of partial-moment differential equations may be required, in order to update E[N p (t)] at the times of the deterministic change. The computational effort spent on approximating the p th moment of N(t), using partial-moment differential equations to update p th moment differential equation of N(t), may potentially be more than the computational effort required to numerically integrate the full set of KFEs. Thus, we approximate the probabilities on the RHS of equation (1.19), using the SDA method. We match the first two moments of the random variable Z t, where Z t = (N(t) N(t) c k 3 k 2 ), to a PE(c k 3 k 2, θ, γ), and approximate all required state probabilities. Method 1 requires approximated state probabilities using the SDA approach. For Method 1, the maximum number of state probabilities approximated is (c+1)/2. The maximum number of probabilities that are approximated using Method 2 and SDA approach for different

37 27 change-pairs, is given in Table 1.1. Observe that, numerically integrating E [N(t), N(t) β] does not result in an advantage. Thus for deterministic changes to Ph(t)/Ph(t)/1/c and Ph(t)/M(t)/s/c queueing models, we use the SDA approach, where we match the first two moments of N(t) to the first two moments of X t PE(c, θ, γ), and approximate state probabilities that are required to update E[N p (t)]. Let deterministic additions to N(t) and deterministic deletion from N(t) be referred to as EA (Entity Addition) and ED (Entity Deletion), respectively. Let increases in queue capacity and decrease in queue capacity be referred to as IC (Increase in Capacity) and DC (Decrease in Capacity), respectively. We henceforth refer to all update equations as Change Equations. The method of approximating deterministic changes using the SDA approach is represented by a simple decision tree, shown in Figure (1.9), where Change Equations 1-7 are given by: Change Equation 1: E[N p (t + ɛ )] = ( ) p p (k) p q E[N q (t ɛ )] q= q + c i=c k+1 (c p (i + k) p )P i (t ɛ ). Change Equation 2: E[N p (t + ɛ )] = c k i= ((i + k) p c p )P i (t ɛ ) + c p. Change Equation 3: k 1 E[N p (t + ɛ )] = δ p, i= k 1 i= P i (t ɛ ) + p q= (i k) p P i (t ɛ ). ( ) p ( k) p q E[N q (t ɛ )] q

38 28 Table 1.1: Number of state probabilities approximated using Method 2 Current Magnitude Next Case Magnitude Relation be- Number of prob- Maximum no. of case tween k1,k2 and abilities to be probabilities to c approximated be approximated Addition k1 Addition k2 k2 c + 1 k1 Addition k1 Addition k2 k2 < c + 1 k1 c k1 k2 + 1 (c 1)/2 Addition k1 Deletion k2 k2 k1 Addition k1 Deletion k2 k2 > k1 k2 k1 (c 1)/2 Addition k1 Decrease k2 k2 c + 1 k1 Addition k1 Decrease k2 k2 < c + 1 k1 c k1 k2 + 1 (c 1)/2 Deletion k1 Addition k2 k2 k1 c k2 + 1 c/2 Deletion k1 Addition k2 k2 < k1 c k1 + 1 (c 1)/2 Deletion k1 Deletion k2 k2 c + 1 k1 k2 (c 1)/2 Deletion k1 Deletion k2 k2 > c + 1 k1 k2 1 (c 1)/2 Deletion k1 Decrease k2 k2 k1 c k2 + 1 c/2 Deletion k1 Decrease k2 k2 < k1 c k1 + 1 (c 1)/2 Decrease k1 Addition k2 N.A. c k1 k1 + 1 (c k1 + 1)/2 Decrease k1 Deletion k2 N.A. k2 (c k1 + 1)/2 Decrease k1 Decrease k2 N.A. k2 (c k1 + 1)/2

39 29 EA k c k + 1 k > c k + 1 Change Equation 1 Change Equation 2 Deterministic Change ED k c k + 1 k > c k + 1 Change Equation 3 Change Equation 4 IC Change Equation 5 DC k c k + 1 k > c k + 1 Change Equation 6 Change Equation 7 Figure 1.9: Simple decision tree for deterministic changes to M(t)/M(t)/1/c queueing model Change Equation 4: k 1 E[N p (t + c ɛ )] = δ p, P i (t ɛ ) + (i k) p P i (t ɛ ). i= i=k Change Equation 5: E[N p (t + ɛ )] = E[N p (t ɛ )]. Change Equation 6: E[N p (t + ɛ )] = E[N p (t ɛ )] + c ((c k) p i p )P i (t ɛ ). i=c k+1

40 3 Change Equation 7: E[N p (t + ɛ )] = c k i= (i p (c k) p )P i (t ɛ ) + (c k) p. 1.3 Algorithm SSMQ Let the time interval of interest be (t, t f ). For each value of time t (t, t f ), assume p 2 and assume the values of E[N q (t )] are known, where q {1,..., p}. Let the solution algorithm be labelled Single Server Markovian Queue (SSMQ). The algorithm to approximate deterministic changes to the M(t)/M(t)/1/c queueing model, using the SDA approach is given by: Algorithm SSMQ for t = t t f do if a deterministic change is to be made at time t then 1. Set E[N q (t )] E[N q (t)], for q {1,..., p} 2. Match E[N(t )] and E[N 2 (t )] to the first two moments of X t PE(c, θ t, γ t ) 3. Evaluate E[N q (t + )] for q {1,..., p}, using E[N q (t )], for q {1,..., p}, and X t as a surrogate for N(t ) 4. Update c if required 5. Set E[N q (t)] E[N q (t + )], for q {1,..., p} else

41 31 1. Match E[N(t)] and E[N 2 (t)] to the first and second moment of X t PE(c, θ, γ), respectively 2. Approximate P (t) and P c (t) using X t as a surrogate for N(t) 3. Numerically integrate E[N q (t)], where q {1,..., p}, to get approximations for E[N q (t + t)] 4. Set t t + t if for. 1.4 Conclusion We presented a technique to approximate the p th moment of N(t), using a mapping scheme for state probabilities and the SDA approach described by Taaffe and Ong ([8],[1]), when deterministic changes are made to the M(t)/M(t)/1/c queueing model. It was shown that approximating state probabilities, to update the value of p th moment of N(t), at the time of a deterministic change, requires less computational effort as compared to numerically integrating partial moment differential equations to do the same. We next study deterministic changes to the Ph(t)/Ph(t)/1/c and Ph(t)/M(t)/s/c queueing models. We use the SDA approach to approximate deterministic change to both queueing models.

42 Chapter 2 Deterministic changes to the Ph(t)/Ph(t)/1/c queueing model The versatility and flexibility of the Phase-type distribution is attributed to its denseness over (, ]. Johnson and Taaffe [2] showed, that a general non-negative continuous distribution can be arbitrarily closely approximated by a Phase-type distribution. Ong and Taaffe ([8]) note, that the Phase-type distribution s flexibility in approximating non-negative continuous distributions can be utilized in approximating a time-varying GI/G/1/c queueing model, with a Ph(t)/Ph(t)/1/c queueing model. The arrival and service process for the Ph(t)/Ph(t)/1/c queueing model are both described by time-varying Phase-type distributions. A stationary Phase-type distribution having m phases can be described by a continuous time Markov process with m transient states and 32

43 33 one absorbing state. An initial-state probability vector α contains the probability of starting in any of the m transient states. The vector λ contains the inverses of the mean times spent in the transient states; i.e. the vector of rates for the underlying Markov process. Given the routing probabilities among the m + 1 states, and given that the process starts in a transient state, the time until absorption random variable has a Phase-type distribution. This random variable is said to have a time-varying Phase-type distribution if the routing probability matrix, α, and λ are all time depent. For the Ph(t)/Ph(t)/1/c queueing model, let the number of phases in the arrival process be m A and let the number of phases in the service process be m B. Referring to Ong and Taaffe([8]), the time-varying phase arrival process and the time-varying phase service process are described by (A(t), λ(t)) and (B(t), µ(t)) respectively. A(t) and B(t) are the underlying Markov chains of the arrival and service process, given by A(t) = A 1 (t) A 2 (t) α(t) B(t) = B 1 (t) B 2 (t) β(t). The matrix A 1 (t) is the routing probability matrix of size m A m A, for transient-to-transient states in the underlying Markov chain for the arrival process. Similarly B 1 (t) is the routing probability matrix of size m B m B, of the transient-to-transient states in the underlying Markov chain for the service process. A 2 (t) and B 2 (t) are vectors, of size m A 1 and m B 1, respectively, representing the probabilities of being absorbed from transient states for their respective processes. The vector λ(t) is the time depent vector of rates for the arrival process, and µ(t) is the vector of rates corresponding to the service process.

44 34 An actual arrival occurs when an absorption takes place in the underlying Markov chain for the arrival process. This is similar to the service process, where an actual completion of service occurs when an absorption takes place in the underlying Markov chain for the service process. Once an absorption occurs, for either arrival or service process, the underlying Markov chain restarts. To complete the description of the arrival and service processes we define an initial-state probability vector for each process. The initial-state probability vector contains the probabilities of starting in any one of the transient states. The vector α(t) is the initial-state probability vector for the arrival process, and the vector β(t) is the initial-state probability vector for the service process, at time t. For the Ph(t)/Ph(t)/1/c queueing model, we define the state variable as a triple (N(t), I(t), J(t)), where N(t) is a random variable that denotes the number of entities in the system at time t, I(t) is a random variable that denotes the current-arrival phase of the next arrival at time t and J(t) is a random variable that denotes the current-service phase of an entity at time t. For the Ph(t)/Ph(t)/1/c queueing model, N(t) {,..., c}, I(t) {1,..., m A }, and J(t) {,..., m B }. The current-service phase at time t, J(t), is equal to zero if the system is idle at time t. The state space Ω = {(, i, ) i {1,..., m A }} {(n, i, j) n {1,..., c}, i {1,..., m A }, j {1,..., m B }} (Refer [8]). In order to approximate the p th moment of N(t) using the SDA approach, Ong and Taaffe ([8]) partitioned the state space. Define the indexing set D = {(i, j) (i, j) {1,..., m A } {1,..., m B }}.

45 35 Define a subspace, Ω i,j, as Ω i,j = {(n, i, j) n {1,..., c}, (i, j) D}. Define a subspace Ω = {(, i, ) i {1,..., m A }}. All the state probabilities Ω will be collectively referred to as idle-state probabilities. The KFEs for the Ph(t)/Ph(t)/1/c queueing model are: dp,i, (t)/dt = λ i (t)p,i, (t) + m B + l=1 m A k=1 b l,mb +1(t)µ l (t)p 1,i,l (t) dp n,i,j (t)/dt = (λ i (t) + µ j (t)) P n,i,j (t) + + m A k=1 +α i (t) a k,i (t)λ k (t)p,k, (t) m B a k,i (t)λ k (t)p n,k,j (t) + b l,j (t)µ l (t)p n,i,l (t) m A k=1 l=1 a k,ma+1(t)λ k (t) (δ n1 β j (t)p,k, (t) +δ n1 {P n 1,k,j (t) + δ nc P c,i,j (t)} ) m B +δ nc β j (t) b l,mb +1(t)µ l (t)p n+1,i,l (t) where (i, j) {1,..., m A }, j {1,..., m B } and n {1,..., c}. l=1 For each Ω i,j, where (i, j) D, define E (p) i,j (t) as E[N p (t), N(t) 1, I(t) = i, J(t) = j]. Thus de (p) i,j (t)/dt = (λ i (t) + µ j (t)) E (p) i,j (t) m A a k,i (t)λ k (t)e (p) m B k,j (t) + + k=1 l=1 m A +α i (t) a k,ma+1(t)λ k (t) β j (t)p,k, (t) k=1 ( ) m A p +α i (t) a k,ma+1(t)λ k (t) p E (q) p 1 k,j k=1 q= q (t) q= b l,j (t)µ l (t)e (p) i,l (t) ( ) p q c q P c,k,j (t)

46 ( ) m B p +β j (t) b l,mb +1(t)µ l (t) p ( 1) p q E (q) i,l l=1 q= q (t) δ p, P 1,i,l (t). (2.1) 36 Let Ω 1 = Ω i,j. (i,j) D Observe that, Ω = Ω Ω 1 and note that E[N p (t)] = or E[N p (t)] = m A m B i=1 j=1 m A m B i=1 j=1 E (p) i,j (t) when p 1 E (p) m A i,j (t) + i=1 P,i, (t) when p = 1 We refer to E (p) i,j (t) as the p th partial-moment differential equation (PMDE) of Ω i,j. Ong and Taaffe ([8]) used the SDA approach to approximate E (p) i,j (t) for each subspace Ω i,j. Numerical integration of equation (2.1), for (i, j) D, and for every q {,..., p}, requires the values of idle-state probabilities, along with the state probabilities P 1,i,j (t) and P c,i,j (t), for (i, j) D, in the time interval of interest. We numerically integrate the KFEs for the idle-state probabilities and approximate the state probabilities P 1,i,j (t) and P c,i,j (t), for (i, j) D, by using the SDA and state-space partitioning approach described by Ong and Taaffe ([8]). For each subspace Ω i,j, define a random variable X ij t PE(c 1, θ ij t, γ ij t ). Match the conditional moment (E (1) i,j (t)/e () i,j (t)) 1 to the first moment of X ij t, and match the conditional

47 37 moment (E (2) i,j (t)/e () i,j (t)) 2((E (1) i,j (t)/e () i,j (t)) 1) 1 to the second moment of X ij t, to determine the parameters θ ij t and γ ij t. The approximate values of P 1,i,j (t) and P c,i,j (t) are given by P[X ij t = ] and P[X ij t = c 1], respectively. We use the SDA and state-space partitioning approach described by Ong and Taaffe ([8]), to approximate deterministic changes to the Ph(t)/Ph(t)/1/c queueing model. To approximate the p th moment of N(t), we update m A idle-state KFEs and (p + 1)m A m B PMDEs, at the times of deterministic changes to the Ph(t)/Ph(t)/1/c queueing model. There are four possible deterministic changes that can be made to the Ph(t)/Ph(t)/1/c queueing model. They are: EA, ED, IC and DC. No deterministic change to the number of servers is allowed for the Ph(t)/Ph(t)/1/c queueing model. We present a mapping scheme for the state probabilities, in Ω and Ω i,j, where (i, j) D. Note, there are many mapping schemes for state probabilities with respect to deterministic changes made to the Ph(t)/Ph(t)/1/c queueing model. The one presented in this thesis represents one set of rules. The mapping scheme presented here is based on the rule that if a deterministic change requires an entity to be cleared from the system then it will be cleared. The system has no memory of any entity that is cleared (counter to this, is the mapping scheme where cleared customers can enter the system at a later time, and thus the system does not lose entities due to a deterministic change).

48 Deterministic addition to N(t) Let z entities be deterministically added to the system at a selected time t ɛ. Let the system capacity at time t ɛ be c. Since the system has finite capacity, if z c, then z = c. At time t ɛ the mapping scheme for the state probabilities is: P,i, (t + ɛ ) = P 1,i,j (t + ɛ ) =. P z,i,j (t + ɛ ) = β j (t + ɛ ) P,i, (t ɛ ) P z+1,i,j (t + ɛ ) = P 1,i,j (t ɛ ). P c 1,i,j (t + ɛ ) = P c z 1,i,j (t ɛ ) P c,i,j (t + ɛ ) = c n=c z P n,i,j (t ɛ ) for i {1,..., m A }, j {1,..., m A } and n {1,..., c}. 2.2 Deterministic deletion from N(t) Deleting entities in the queue is possible only when N(t) 1. Let z entities be deterministically deleted from the system at a selected time t ɛ. Let the system capacity at time t ɛ be c. Since there can be at most c entities in the queue, if z c, then z = c. At time t ɛ the

49 39 mapping scheme for the state probabilities is: P c,i,j (t + ɛ ) = P c z+1,i,j (t + ɛ ) = P c z,i,j (t + ɛ ) = P c,i,j (t ɛ ).. P 1,i,j (t + ɛ ) = P z+1,i,j (t ɛ ) P,i, (t + ɛ ) = P,i, (t m B z ɛ ) + P n,i,j (t ɛ ) j=1 n=1 for i {1,..., m A }, j {1,..., m A } and n {1,..., c}. 2.3 Deterministic increase in queue capacity c 1 Let the system capacity at time t ɛ be c. Given that the system capacity is c, let the queue capacity be deterministically increased by z at time t ɛ. The mapping scheme for the state probabilities at time t ɛ is: P,i, (t + ɛ ) = P,i, (t ɛ ) P 1,i,j (t + ɛ ) = P 1,i,j (t ɛ ). P c,i,j (t + ɛ ) = P c,i,j (t ɛ )

50 4 P c+1,i,j (t + ɛ ) = P c+z,i,j (t + ɛ ) =. for i {1,..., m A }, j {1,..., m A } and n {1,..., c}. 2.4 Deterministic decrease in queue capacity c 1 Deterministic decrease in queue capacity reduces the capacity of the system to queue potential entities. If reduction in queue capacity requires deletion of entities from the system, then entities are deleted from the system. Let the system capacity at time t ɛ be c. Given that the system capacity is c, let the queue capacity be reduced by z at time t ɛ. The mapping scheme for the state probabilities is: P c,i,j (t + ɛ ) =. P c z+1,i,j (t + ɛ ) = P c z,i,j (t + ɛ ) = P c z,i,j (t ɛ ) + c P n,i,j (t ɛ ) P c z 1,i,j (t + ɛ ) = P c z 1,i,j (t ɛ ). P 1,i,j (t + ɛ ) = P 1,i,j (t ɛ ) n=c z+1

51 41 P,i, (t + ɛ ) = P,i, (t ɛ ) for i {1,..., m A }, j {1,..., m A } and n {1,..., c}. 2.5 Updating idle state KFEs and PMDEs Using the mapping scheme presented in the last four sections, for the state probabilities, the solution technique for updating the idle state KFEs and all PMDEs, given that a deterministic change is made at time t ɛ, is represented by the decision tree given in Figure (2.1), where Change Equations 1-7 are given by Change Equation 1: For i {1,..., m A } P,i, (t + ɛ ) = (2.2) and for (i, j) D E (p) i,j (t + ɛ ) = z p β j (t + ɛ )P,i, (t ɛ ) + + c n=c z+1 p q= ( ) p z p q E (q) i,j (t ɛ ) q {c p (n + z) p }P n,i,j (t ɛ ). (2.3) Change Equation 2: For i {1,..., m A } P,i, (t + ɛ ) = (2.4)

52 42 EA z c z z > c z Change Equation 1 Change Equation 2 Deterministic Change ED z c z z > c z Change Equation 3 Change Equation 4 IC Change Equation 5 DC z c z z > c z Change Equation 6 Change Equation 7 Figure 2.1: Decision tree for deterministic changes to Ph(t)/Ph(t)/1/c queueing model and for (i, j) D E (p) c z i,j (t + ɛ ) = z p β j (t + ɛ )P,i, (t ɛ ) + {(n + z) p c p }P n,i,j (t ɛ ) + c p E () i,j (t ɛ ). (2.5) n=1 Change Equation 3: For i {1,..., m A } P,i, (t + ɛ ) = P,i, (t m B z ɛ ) + P n,i,j (t ɛ ) (2.6) j=1 n=1 and for (i, j) D E (p) i,j (t + ɛ ) = p q= ( ) p ( z) p q E (q) i,j (t ɛ ) q

53 43 z (n z) p P n,i,j (t ɛ ). (2.7) n=1 Change Equation 4: For i {1,..., m A } P,i, (t + ɛ ) = P,i, (t m B ɛ ) + E () i,j (t ɛ ) j=1 c n=z+1 P n,i,j (t ɛ ). (2.8) and for (i, j) D E (p) i,j (t + ɛ ) = c n=z+1 (n z) p P n,i,j (t ɛ ). (2.9) Change Equation 5: For i {1,..., m A } P,i, (t + ɛ ) = P,i, (t ɛ ) (2.1) and for (i, j) D E (p) i,j (t + ɛ ) = E (p) i,j (t ɛ ). (2.11) Change Equation 6: For i {1,..., m A } P,i, (t + ɛ ) = P,i, (t ɛ ) (2.12) and for (i, j) D E (p) i,j (t + ɛ ) = E (p) i,j (t ɛ ) + c n=c z+1 {(c z) p n p }P n,i,j (t ɛ ). (2.13)

54 44 Change Equation 7: For i {1,..., m A } P,i, (t + ɛ ) = P,i, (t ɛ ) (2.14) and for (i, j) D E (p) i,j (t + ɛ ) = (c z) p E () c z i,j (t ɛ ) + {n p (c z) p }P n,i,j (t ɛ ). (2.15) n=1 2.6 Algorithm SSFC Let the solution algorithm for deterministic change in the Ph(t)/Ph(t)/1/c system be labelled as SSFC (Single Server Finite Capacity) algorithm. Let the interval of interest be (t, t f ) and p 2. Assume, we know the values of P,i, (t ), where i {1,..., m A }, and E (q) i,j (t ), where j {1,..., m B } and q {,..., p}. Algorithm SSFC for t = t t f do if a deterministic change is to be made at t then 1. For each Ω i,j, match the conditional first moment and the conditional second moment (E (1) i,j (t ɛ )/E () i,j (t ɛ )) 1 (E (2) i,j (t ɛ )/E () i,j (t ɛ )) 2(E (1) i,j (t ɛ )/E () i,j (t ɛ )) 1

55 45 to the first and second moments of X ij t PE(c 1, θ ij t, γ ij t ), respectively, and approximate all required state probabilities 2. Set E () i,j (t ɛ ) E () i,j (t ɛ ), E (1) i,j (t ɛ ) E (1) i,j (t ɛ ), E (2) i,j (t ɛ ) E (2) i,j (t ɛ ), for (i, j) D, and set P,i, (t ɛ ) P,i, (t ɛ ), for i {1,..., m A } 3. Use approximated state probability values, along with values of E () i,j (t ɛ ), E (1) i,j (t ɛ ), and E (2) i,j (t ɛ ), to approximate E () i,j (t + ɛ ), E (1) i,j (t + ɛ ), E (2) i,j (t + ɛ ), for (i, j) D, and P,i, (t + ɛ ), for i {1,..., m A } 4. Set E () i,j (t ɛ ) E () i,j (t + ɛ ), E (1) i,j (t ɛ ) E (1) i,j (t + ɛ ), E (2) i,j (t ɛ ) E (2) i,j (t + ɛ ), for (i, j) D and P,i, (t ɛ ) P,i, (t + ɛ ) for each i {1,..., m A } 5. Update c if required else 1. For each Ω i,j, match the conditional first moment (E (1) i,j (t ɛ )/E () i,j (t ɛ )) 1 and the conditional second moment (E (2) i,j (t ɛ )/E () i,j (t ɛ )) 2(E (1) i,j (t ɛ )/E () i,j (t ɛ )) 1 to the first and second moments of X ij t PE(c 1, θ ij t, γ ij t ), respectively 2. For each Ω i,j, approximate P 1,i,j (t) and P c,i,j (t) 3. Use the approximated state probability values to numerically integrate PMDEs and idle state KFEs from t t + t, to get approximations for P,i, (t + t), for each i {1,..., m A }, and E () i,j (t + t), E (1) i,j (t + t), E (2) i,j (t + t), for (i, j) D

56 46 4. Set t t + t if for. 2.7 Conclusion We used the state-space partitioning and the SDA approach described by Ong and Taaffe [8] to approximate deterministic changes to the Ph(t)/Ph(t)/1/c queueing model. The solution technique along with the Change Equations presented, minimize the number of approximated state probabilities.

57 Chapter 3 Deterministic changes to the Ph(t)/M(t)/s/c queueing model The Ph(t)/Ph(t)/s/c queueing model is a flexible queueing model. The property of the Phase-type distribution to be able to approximate any non-negative continuous distribution, arbitrarily closely, ensures that the Ph(t)/Ph(t)/s/c queueing model can be used to approximate time-varying queueing models, where the time between arrivals and service time are described by non-negative continuous distributions. The Ph(t)/M(t)/s/c queueing model is a special case of the Ph(t)/Ph(t)/s/c queueing model, where the service time has a timevarying Phase-type distribution with one phase, and rate µ(t). The time between arrivals is represented by a time-varying Phase-type distribution. Let the maximum number of arrival phases be m A. The arrival process is described by (A(t), λ(t)), where A(t) represents the underlying Markov chain for the arrival process, and as defined in Chapter 2 is given by 47

58 48 A(t) = A 1 (t) A 2 (t) α(t). The vector λ(t) is the vector of rates for the arrival process. At the start of the time interval of interest, there are s servers, and the queue capacity is c s. Unlike the Ph(t)/Ph(t)/1/c queueing model, deterministic changes to the number of servers is allowed for the Ph(t)/M(t)/s/c queueing model. We define the state variable as a pair (N(t), I(t)), where N(t) is a random variable that denotes the number of entities in the system at time t, and I(t) is a random variable that denotes the current phase of the next arrival at time t. For the Ph(t)/M(t)/s/c queueing model, N(t) {,..., c} and I(t) {1,..., m A }. The state space Ω = {(m, i) m {,..., c}, i {1,..., m A }}. Let P m,i (t) denote P[N(t) = m, I(t) = i]. The KFEs for the Ph(t)/M(t)/s/c queueing model are: dp n,i (t)/dt = ( λ i (t) + δ n, nµ(t) ) P n,i (t) + a j,i (t)λ j (t)p n,j (t) m A m A j=1 +δ n, α i (t) a j,ma +1 λ j (t)p n 1,j (t) j=1 +(n + 1)µ(t)P n+1,i (t) dp k,i (t)/dt = m A (λ i (t) + sµ(t)) P k,i (t) + a j,i (t)λ j (t)p k,j (t) m A j=1 +α i (t) a j,ma +1 λ j (t) (P k 1,j (t) + δ n,c P c,j (t)) j=1 +δ n,c sµ(t)p k+1,i (t) where n {,..., s 1}, k {s,..., c} and i {1,..., m A }.

59 49 Taaffe and Ong ([1]) partitioned the state space into subspaces to approximate the p th moment of N(t). Define Ω 1,i = {(n, i) n {,..., s 1}, i {1,..., m A }} and Ω 2,i = {(k, i) k {s,..., c}, i {1,..., m A }}. Let and Ω 1 = i {1,...,m A } Ω 1,i Ω 2 = Ω 2,i. i {1,...,m A } Let E (p) i,1 (t) denote E[N p (t), N(t) s 1, I(t) = i] and E (p) i,2 (t) denote E[N p (t), N(t) s, I(t) = i]. We refer to de (p) i,1 (t)/dt as the p th PMDE of Ω 1,i and refer to de (p) i,2 (t)/dt as the p th PMDE of Ω 2,i. The p th PMDE of Ω 1,i and Ω 2,i are given by: de (p) i,1 (t)/dt = λ i (t)e (p) m A i,1 (t) + a j,i (t)λ j (t)e (p) j,1(t) j=1 ( ) m A p +α i (t) a j,ma +1(t)λ j (t) p E (q) j=1 q= q j,1(t) s p P s 1,j (t) p 1 ( ) +µ(t) p ( 1) p q E (q+1) i,1 (t) + (s 1) p s P s,i (t) (3.1) q= q

60 5 de (p) i,2 (t)/dt = λ i (t)e (p) m A i,2 (t) + a j,i (t)λ j (t)e (p) j,2(t) j=1 ( ) m A p +α i (t) a j,ma +1(t)λ j (t) p E (q) p 1 ( ) p j=1 q= q j,2(t) c q P c, j(t) q= q m A +α i (t) a j,ma +1(t)λ j (t)s p P s 1,j (t) j=1 p 1 ( ) +sµ(t) p ( 1) p q E (q) i,2 (t) (s 1) p P s,i (t) q= q (3.2) where i {1,..., m A }. Observe that Ω = Ω 1 Ω 2 and note that E[N p (t)] = m A (E (p) i=1 i,1 (t) + E (p) i,2 (t)). Taaffe and Ong ([1]) used the SDA approach to approximate the p th PMDE of Ω 1,i, and the p th PMDE of Ω 2,i, for i {1,..., m A }. Numerical integration of the p th PMDE of Ω 1,i and Ω 2,i requires the values of P s 1,i (t), P s,i (t), and P c,i (t), for i {1,..., m A }, in the time interval of interest. To approximate P s 1,i (t), for i {1,..., m A }, define a random variable Xt 1i PE(s 1, θt 1i, γt 1i ). Match the conditional moment E (1) i,1 (t)/e () i,1 (t) to the first moment of X 1i t and the conditional moment E (2) i,1 (t)/e () i,1 (t) to the second moment of Xt 1i. Once θt 1i and γ 1i t are determined, the approximate value of P s 1,i (t) is given by P[X 1i t = s 1]. To approximate P s,i (t) and P c,i (t), for i {1,..., m A }, define a random variable Yt 2i PE(c s, θt 2i, γt 2i ). Match the conditional moment (E (1) i,2 (t)/e () i,2 (t)) s