Digitized by the Internet Archive in 2013 http://archive.org/details/timedependentdes579barr UIUCDCS-R-73-579 yyuZi June 1973 TIME -DEPENDENT DESCRIPTORS FOR THE POISSON QUEUE by RHODA HORNKOHL BARR THE LIBRARY OF THE inn 9 1973 Report No. UIUCDCS-R-73-579 TIME-DEPENDENT DESCRIPTORS FOR THE POISSON QUEUE BY RHODA HORNKOHL BARR June 1973 Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 6l801 *This work was supported in part by contract No. NSF GJ 28289 and was submitted in partial fulfillment of the requirements for the Master of Science degree in Computer Science, June 1973 < 11 ACKNOWLEDGEMENT The author would like to express her sincere gratitude to her advisor, Edward K. Bowdon, Sr. , without whose advice, criticism, and patience under less than ideal circumstances this work could not have he en completed. The author is also indebted to Mrs. Gayanne Carpenter for her invaluable ability to cut red tape. Finally, special thanks are due my husband, Bill, who insisted on putting me through school. Ill TABLE OF CONTENTS Page 1 . INTRODUCTION 1 2. TRANSIENT SOLUTIONS TO THE POISSON QUEUE 9 3 . LAPLACE TRANSFORMS OF TRANSIENT SYSTEM DESCRIPTORS ll* h . NUMERICAL INVERSION OF THE LAPLACE TRANSFORMS 21 5. CONCLUSION , 33 APPENDICES 35 LIST OF REFERENCES 6j 1 . INTRODUCTION Man has "been searching for ways to avoid waiting in line for thousands of years. In ancient Rome the man who didn't want to wait all day in the "breadline elbowed his way to the head of the line. Today the modern shopper confronted with six checkout counters tries to guess which line will move most quickly. The appli- cation of analytical tools such as modeling and simulation to queueing problems represents a small triumph for civilization, as we replace force and guesswork by reason in our attempts to reduce waiting times. One of the first applications of queueing theory was to the problem of sched- uling appointments with physicians so that patients" waiting time is kept to a minimum while physicians are kept busy [8, 10]. Other obvious applications have been to customer queues in banks and supermarkets. The advent of third-generation computers, however, has provided the major impetus for research in this area. The reason for this is hardly obscure. Few human beings can consistently command more than $100.00 per hour for their services. While accounting methods vary, the use of a large- or medium-scale computing system costs on the order of $1.00 per second. Thus a solution to the problem of keeping a computer from waiting idle for its next task becomes economically, if not aesthetically, more important than problems related to the convenience of people who don't like to wait in lines. Happily, results derived from the study of jobs waiting for service from a computer are also applicable to the study of customers waiting for a doctor or bank teller, because queueing analysis is largely independent of any particular application. The first step in analyzing a queueing system is usually to deduce the nature of arrivals and service in the system, perhaps make some simplifying assumptions about it and then construct a mathematical model of the system. In this paper we are concerned with Poisson queueing systems which are schematized in Figure 1 and have the following characteristics: Input Population ooo Queue Servers The k-Server Poisson Queueing System Figure 1 1. infinite arrival population 2. exponentially distributed arrivals with average arrival rate A > 3. k homogeneous servers in parallel k. service discipline is "first-come, first-served" 5. exponentially distributed service with average service rate y > 6. infinite queue length 7. initially there are i units in the system awaiting service, i > 0. For this system it is known that the probability of n arrivals in a time interval of length At is iM|£ e ^At ^ ^ probabmty Qf q services by a single server in an interval of the same duration is &*&&! e ~^t *„ We denote by P n (t) the probability that at a given "time t there are exactly n units in the system, both waiting and in service. Where there is one server the system has been described by the following set of differential difference equations: p o (t) = - AP o (t ) + nV t} (1.1) P>) = -(X + y)P (t) + AP n _ l( t) + yP n+1 (t), n > 1. (1 . 2) The differential difference equations which describe the system with k > 1 servers are: PqU) = -AP Q (t) + yPi ( t ) (l<3) i P n (t) = -( X+ ny)P n (t) + AP n _ 1 (t) + (n+DyP^t ) , 1 < n < k (1.4) P n (t) = -(A + ky)P n (t) + AP^t) + kyP n+1 (t), n > k. (l . 5 ) Some specific systems which can be described by this model are single-processor imputing systems such as the Univac 1106 or multi-processor systems which select tasks from a single queue, such as the Univac 1108 and Univac 1110. Some localized Ij network computers with negligible transmission times, such as the Distributed Computer System [5] and the Collins C-System [3], can also be treated as multi- processor systems. Another application for the model is to systems which reach one steady state but don't remain in it. A change in arrival rate or the number of servers, for example, would cause a transition from one steady state to another. The model could be used to describe this transition. Having constructed a mathematical model of a queueing system, one would then like to use it to find out how the system operates. Some of the questions one :,.ight ask are: How many units are likely to be waiting for service at a given time? How often will there be no units at all waiting to be served? How often will the servers be left idle? With what probability will there be exactly n units in the system (i.e., what is P^t))? Does the system ever reach a point in time after which the values P (t) are static? When the answer to this last questions is yes, the system is said to reach steady state. This is known to occur if and only if A < ku. Since the system is no longer changing with respect to time, P^t) = 0. Moreover, as t ■+ -, P^t) takes on the static value that no longer depends on t, denoted P q . The well- known solution to the steady state problem is when there is one server, and P = ^rY^Vn' < n < k (1.7) n n! VV / ° P = T-r-t P n> 1J > k (1 ' 8) n k n_k k! W ° Ik-l '- 1 i) J- At + -1- A\* JSli_ )/_, n! W k! W ky-A ( (l.9) n=0 where there are k > 1 servers. Many other descriptors have been determined for the queueing system at steady state. The expected number of units in the system is defined as n=0 In the one-server case this is evaluated as — - . In the many-server case it is y-A evaluated as U/y) k AuP n . U + - • (1.11) (k-l)!(ky-A) 2 y Sometimes one is interested only in the number of units in the queue rather than the number of units in the system, which includes both units in service and those waiting in the queue. The probability of having exactly m units in the queue at steady state, denoted P , is defined as mq k P (1.12) n=0 P = P _,_ , m > 1. (1.13) mq m+k - w-.jo; Then the expected number of units in the queue is defined as E = ) mP m /_, mq m=0 d.iM In the one-server case this is evaluated as U/y) r-> -,c\ and in the many-server case it is U/y) k AyP 2 — . (1.16) (k-l)!(ky-A) 2 When studying a queueing system with more than one server it may he useful to know something about how often the servers are idle. If exactly n servers are idle then there are k - n units in the system and the expected number of idle servers is defined as k E T = ) nP, I [_> k-n n=0 or equivalently k-1 V (k-n)P n . (LIT) n=0 This is evaluated as k - — [12]. Thus at steady state we know how many units to expect in the queue and in the system (both waiting in queue and in service), and how many servers will be idle. A steady state analysis is most useful when the system is known to reach steady state quickly and outside interference will not cause it to be jolted out of steady state. A sensitivity analysis of the Poisson queueing system shows that the smaller the ratio A/ky the faster the system reaches steady state [l^]. For such systems a transient analysis supplies little extra information, but only the transient analysis can show precisely how fast the system reaches steady state and 7 that it actually stays there. Moreover, a serious shortcoming of steady state analysis is that it cannot "be applied at all to the many systems which never reach steady state. For these systems only a time- dependent analysis is possible. Many time-dependent, or transient, solutions to the differential difference equations modelling the Poisson queueing system have "been discussed in the litera- 3P (t) ture. First Bailey [l] determined an exact analytic expression for —r- in the at one-server case. Later Saaty [12] and Cox and Smith [h] determined P (t) in the n one-server case, and Saaty [13] derived an expression for the Laplace transform of P (t) in the many-server case. Unfortunately, all of these expressions are far more complicated than the steady state solutions and it is difficult to obtain really useful information from them. This problem will be discussed further in Chapter II. Several other descriptors for the system at steady state were given above. Just as at steady state, one could ask how many units in the system, units in the queue, or idle servers can be expected at a given time. We define time- dependent functions which are analogous to these steady state descriptors. The expected number of units in the system at time t is E n (t) = V tfjt) n=0 (1.18! The probability of having m units in the queue at time t is n=0 % (t) " W*>- (1-20) Then the expected number of units in the queue at time t is E (t) = ) mP (t) m /_, mq (1.21) mq m=0 Finally, the expected number of idle servers at time t when there are more than one server is k-1 E (t) = Y (k-n)P n (t). n=0 In Chapter III we determine analytically the Laplace transforms of the functions Ejt), Ejt) and E (t) for the cases k = 1 and k = 2. We n m further show how our method can be applied to achieve similar results when k > 2. In Chapter IV these functions are inverted numerically for some selected values of the parameters X,y, k and i. We produce tables of values for E^t), E^t) and E^t) in these cases. Finally, in Chapter V we discuss some implications of this work and suggest areas for future research. (1.22) 2. TRANSIENT SOLUTIONS TO THE POISSON QUEUE In this chapter we discuss some solutions for P (t) in the one-server and many- server cases. We are concerned mainly -with preparing a foundation for the work presented in Chapter III. Saaty's [12] complete solution to the single server Poisson queue is outlined in Appendix I . He finds that p B c*>-.-^*i($rw^rw OO j| j=n+i+2 where I. is the modified Bessel function of the first kind with argument 2\/Xy t. Many other solutions have "been given for this problem. Bailey [l] shows that 3P (t) / f~4.+n ,, , N , ( + x/Xy I. ._ + XI. . _ - 2\J^j I + yl CP p n v i-n+1 i+n+2 i+n+1 i+nj ^.^J where I. is the modified Bessel function of the first kind with argument 2V^y t. Cox and Smith [h] formulate the problem somewhat differently. The time unit is chosen so that the parameter of the service distribution y = 1. The corres- ■ ponding parameter for the arrival distribution is p. Then for the one server case the differential difference equations become PqU) = - P P (t) + P (t) ( 2 .3) 10 P n (t) = PP n-1 (t) - (l+p)P n (t) + P n+1 (t), n > 0. (2.U) Now it is assumed that (2.k) holds for all integers n. If a solution can then be found satisfying the additional constraint P n (t) = pP , (t), (2.3) is satis- fied and it is the required solution- It is found to he P (t) = P ^n-D e -(1+P)t h-n+fi ^i+l -p) > p - h ~ h h n+i+l+j 3=1 (2.5) This is completely analogous to the solution given by Saaty. It can be proved [12] that P (t) = 1, n n=0 (2.6) This is a stronger result than the steady state solution. As shown in Appendix I in the transient solution Rouche's theorem is used to determine P (s). But at steady state it must be assumed that P = 1 n (2.7) n=0 to determine P and complete the solution. It can also be proved [12] that when X < \i ] im P (t) - P . t-*- 00 n n (2.8) Thus the functions P (t) describe the approach of the system to its steady state. n When A • u, units arrive in the system faster than they can be serviced and the 11 length of the queue increases without bound. This is "why no steady state analysis can he made when A > y. In this case, the functions P (t) describe the indefinite n growth of the system [k] . Theoretically the functions P (t) could be used to determine other properties of the Poisson queueing system. In practice the task is formidable. Each function is an infinite sum and each term in the sum includes a Bessel function, which is defined as an infinite sum in Appendix I. Thus the usefulness of these expres- sions for P (t) is limited, n Next we sketch the solution given by Saaty [13]' for the many-server Poisson queue. The differential difference equations which model this problem are given by (1.3), (l.^O, and (1.5). As in the one-server case, we multiply each equation n+1 by z , sum the resulting equations to get an expression in terms of the gener- ating function, take Laplace transforms, and impose the initial condition that P (0) = fi._. We then find n i0 k-1 z 1+1 - y(l-z) V (k-n)zV(s) * If^s)= sz - (l-z^ky-Az) • ■ < 2 "9> It can be shown that |z| < 1 and Re s > are sufficient conditions for convergence of II (z,s) in the k-server case using the same proof as that for the single server case in Appendix I. Then Rouche's theorem can be applied to determine that k-1 i+1 n*/ n a 2 (2.10) V (k-n)aV(s) = -rf r /_, 2 n y(l-a ) n=0 ^ where A+ky+s+V( A+ky+s ) -UkAy a l "' 2A 12 and ^ A+ky+s-V(A+ky+s) -UkAy a 2 = 2A ' (2' 11 ) It is shown that for n > k P*(s) = t i-r- {(A+s)P*(s)[-^- - —J - AP*(s)f-~ - - -~r n (a, -a,JA l I n nl 1 n-1 n-1 \ 2 a l/ \ a 2 a i 1 n-i n-i va 2 a (2.12) * To solve for P (s) when n < k - 1 requires the Laplace transforms of (1.3) and (1.4). They are (A+s)P Q (s) - yP x (s) = & ±o (2.13) (A+s+ny)P (s) - AP _(s) - (n+l)yP _(s) = 6. , 1 < n < k. (2.l4) n n-1 n+1 m In [13] the problem is restricted so that i > k - 1 , which implies that 6. =0. 10 Then for k = 2 (2.10) and (2.13) are solved simultaneously, giving a 2 i+1 P„ ~ (l-a 2 )(2y+sa 2 +Aa 2 ) and # (s+A)a, P. (s) : 777 Uo _, 2 rr P • (2.15) 1 p(l-a M2p+sa 2 +Aa ) L3 For k > 3, (2.10), (2.13), and the first k - 2 equations of (2.11*) can be solved simultaneously to determine P (s), < n < k - 1. For the case k - 2, we extend the results given to hold for any i and find that P * (sl "a* 1 + i io a 2 (1 -»2 ) (T ' (l-a )(2u+sa +AaJ and * (A+s)a2 +1 - 2y6. (l-a 2 ) P l (s) = y(l-a 2 )(2y+sa 2 +Aa 2 )~" ' (2 « l6 ) The solution to the many-server queue is known only in terms of its Laplace transform. The final value theorem for Laplace transforms, which states that lim *, » lim _,, N r , s+0 Sf (s) = t-*- f(t) [11] (2.17) can he used to prove that when A < ku lim _. / , \ _ , t- P n (t) = V (2- 18 ) Thus when A < ky the functions P (s) in some sense describe the system's approach to steady state. All of these transient solutions to the Poisson queue are important because they describe the system at any point in time, and they provide information about systems which don't reach steady state. We would next like to use the results of solving the differential difference equations modelling the Poisson queue to determine some time- dependent descriptors for Poisson queueing systems. Because of the complexity of the functions P (t) and P (s) which have been determined, we n n ' choose to avoid using them directly by concentrating instead on some of Saaty's intermediate results in finding P (t) and P (s). n n Ik 3. LAPLACE TRANSFORMS OF TRANSIENT SYSTEM DESCRIPTORS In this chapter we determine analytically the Laplace transforms of the transient system descriptors E (t ) , E (t ) and E (t) which were defined in the introduction. We begin with the expected number in the system, which includes both units in service and those waiting in the queue. This is given by (l.l8). The straight-forward manner of evaluating this would be to substitute an expression for P (t) from Chapter II. But P (t ) was shown to be a double n n infinite sum, so E (t) would be a triple infinite sum which we have no desire to evaluate. Instead, we recall that the generating function of P (t) is defined in Appendix I as n(*,t) = > z p (t). (3.!) n n=0 Formally, we can say that E (t) =^- (ll(z,t)} n 9z (3.2) z=l This is of limited value because we have no nice expression for Il(z,t). But we do know from equations (i.U) and. (I. 5) of Appendix I that in the one-server case t-t* z " " (!- z ) u o /(l-O IT U.b) - m U y ) - (3 ' 3) W sz - (l-zKy-Az) If we can prove that the proper convergence conditions hold, we will be able to conclude that E*(s) = -~ (Tl*(z,s)} 11 oZ z=l' Now. 15 E n (s) = E (t)e St dt n J_ {n(z,t)> 3z z=l e dt (3.M Also, -- {n (z,s)} z=l 9z n(z,t)e st dt z=l' (3.5) We can conclude that E n (s) =^{n (z,s)} 5=1 (3.6) if 9z (n(z,t)} z=l -St.,, e dt is uniformly convergent, because then 3z (n(z,t)} ~st,, 3 e dt = — z=l 3z n(z,t)e St dt z=l (3.7) We do this by establishing an upper bound E (t). Because arrivals obey a Poisson distribution, the probability of j arrivals in a time interval of length t -'. ' e . We only consider the case where t is finite. If t increases wi th- is out bound either X < ky and one can apply steady state analysis, or A > ky and the expected number in the system increases without bound also. When t is finite, the probability that there are an infinite number of arrivals in the interval 16 (0 t) is lim ' e~ Xt = 0. Thus there can be only a finite number of arrivals, say j of them where j > 1, in (0,t). Since there are i units waiting initially in the system E (t) < i + j . Then ■J- (nU,t)} dZ 7, = 1 e dt = E (t)e St dt < n /. . N -st,, 1 (i+j)e dt = l+J (3.8) which is everywhere uniformly convergent. Then by Weierstrauss ' M-test for integrals [?] dz {Il(z,t)l z=l e dt is also uniformly convergent We can now evaluate E (s) for Ee s > as n ~- {n*(z,s)} 9z z=l 1+1 1+1 - (l-z)ap/(l-a 2 ) dz) sz - (l-z)(y-Az) z=l i+1 S 2 s(l-aj (3.9) The first two terms can be inverted analytically [ll] to determine i+1 s (t) = i + (x-u)t + & H^; where <£~^ indicates the inverse of the Laplace transform. We see that the expected number of units in the system depends on i, the number of units in the system initially when t = 0, increases proportionally to X, the rate at which units arrive in the system, and decreases proportionally to y, the rate at which units are serviced and leave the system. The last term can be considered a correction term which keeps E (t) positive when A < y. When there is more than one server, the proof given above suffices to show that <(.)-■£ {n*(.,.)} Z=l" From Saaty's solution to the many-server Poisson queue which was discussed in Chapter II we know that k-1 i+1 z + y(l-z) V (k-n)zV(s) sz - (l-z) (ky-Az) In the two-server case we can substitute the expressions determined for P n (s) and P 1 (s) and evaluate the partial derivative of IT (z,s) at z = 1 to find ' E * (s) , i + J^H + JSimi} ■ jl_ _ ^io' 1 -^ . (3.,,, n' s g 2 (sa 2 +Aa 2 +2y) s(l-a 2 ) s (scy-Aa 2 +2y ) ^-^ The Laplace inverse of the first two terms [11 ] is i + (A-2y)t. As in the one- server case this shows that the expected number of units in the system depends on i, the number of units waiting in the system initially, increases proportionally to A, the rate at which units arrive in the system, and decreases proportionally to 2y, the rate at which units are serviced by the two servers and leave the system. To determine E (s) when k > 3 is rather more complicated algebraically. First, one must solve k equations in the unknowns P(s),0 3 E (s) can "be determined once expressions for P (s), < m < k - 1, and m * m - - # E (s) are known. , %. For any k, the first two terms in the expression for E (s) can be inverted m analytically to give i - k + (X-ky)t. This shows that when i > k the expected number in the queue depends on the number of units waiting initially in the queue at time t = when the first k units go into service. Further, E (t), like E (t ) , m n increases proportionally to the arrival rate and decreases proportionally to the rate at which units are serviced by the k servers. The final descriptor which we develop here is the expected number of idle servers in the many-server problem, which is given by (1.22). Because this is a finite sume and P (t) < 1 when t is finite, the linearity of the Laplace trans - n forms implies directly that k-1 E I (s) = ) (k-n)P n (s). (3.16) n=0 * This expression can be evaluated once the functions P (s), < n < k - 1, are determined as described in Chapter II. When there are two servers, we find w (2u+A+s)a 2 + <5 i0 E I (s) = y(l-a 2 )(2u+sa 2 +Aa 2 ) " 2y+sa 2 +Aa 2 ' (3 ' 1T) The final value theorem for Laplace transforms [ll] can be used to prove that when \ < kp the functions E (t), E (t), and E T (t) are convergent and approach the n m I 20 steady state values E , E , and E T as t -> <*». This is done explicitly for the n m I case k = 1 and k = 2 in Appendix II . We have determined here expressions for the Laplace transforms of three commonly used system descriptors. None of our expressions is particularly elegant, especially when we recall that a^ is given by A+ky+ s-V( A+ku+s) -UkAy 2A Even if we could invert the Laplace transforms analytically to determine E Q (t), E (t), and E (t), it would be too much to hope that they would be any less compli- m I cated than the expressions for P (t)'. We can, however, invert the Laplace trans- forms numerically and evaluate these transient descriptors at various points in time. This is done in Chapter IV. 21 k. NUMERICAL INVERSION OF THE LAPLACE TRANSFORM In this chapter we determine numerically the inverse Laplace transforms of * * * the functions E n (s), E m (s) and E (s) which were derived in Chapter 3. Many algorithms to do this have been described in the literature; the method used here is that of Weeks [15]. The inverse transform is obtained as a series expansion of Laguerre functions, where the coefficients of the expansion are found by trigo- nometric interpolation. The recurrence relations and quadrature formulae deter- mined by Weeks are used. This method requires a subroutine to evaluate the real and imaginary parts of each function to be inverted. Weeks states that the best results with his method were obtained using a degree of quadrature N between 20 and 50. We have chosen to take N = 20. The rightmost singularity of the functions we are inverting here is at s = 0, so the scale factor c is 0. For most cases, we have chosen to invert functions in the interval < t < 50; then T is 50 and the associated scale factor T is 2.5. All routines were programmed in Fortran IV on an IBM-370, and all floating point arithmetic was performed with double precision. The routine which does the inversion was tested by duplicating Weeks' first example. Considerable round-off error is introduced at three main stages of the compu- tation. First, to calculate the expansion coefficients requires one sine and one 2 9 cosine operation, 2N + 6N + 5 additions and subtractions, 3N + 8N + 11 multipli- cations, 3N + 8 divisions, and N + 1 calls to the f (s) subroutine, where N is the degree of quadrature of the inversion. Once these are known, it requires 2N + l additions and subtractions, to + 5 multiplications and divisions, and two exponen- tiations to evaluate f(t) for a given t. The third problem is the subroutine to evaluate f (s). In testing the inversion routine our results agreed with those of Weeks to within five significant decimal digits. However, the subroutine to evaluate f (s) 22 in this test required only four additions, one subtraction six multiplications, and two divisions. The subroutine to evaluate the functions with which we are concerned here require significantly more computation. Each subroutine must first calculate the real and imaginary parts of a g and then raise a to the (i+l)st power, where i is the number of units waiting initially in the system. Computing a 2 requires eight additions and subtractions, fourteen multiplications and divi- sions, and three square root operations. If i > 1, the exponentiation of the complex number a 2 programmed with real arithmetic requires i 2 + ki + k multipli- cations and i + 2 additions and subtractions. To finally compute the functional value f (s) requires a minimum of fifteen additions and subtractions, twenty multiplications, and two divisions, and a maximum of 33 additions and subtractions, 66 multiplications, and two divisions. After performing all of these arithmetic operations, we are left with at least two but at most three significant decimal digits in the final result f(t). We feel that for many applications these rather imprecise results would be suffi- cient, particularly if the time unit is chosen with care. The order in which calculations were done was found to be critical. For example, to invert E n (s ) when there is one server, one approach would be to invert the term a 2 /s(l-a 2 ) numerically and add to that quantity the inverses of the terms i/s and (X-y)/s , which are easily determined analytically. However, the computed value of ^£ ' o^ /s(l-a 2 )( has only two or three significant digits, and adding to it the quantity i + (X-y)t yields a result of the correct order of magnitude but with no significant digits left. Thus we determine empirically that finite-precision computers do not always respect the linearity of Laplace trans- forms, and it is necessary to invert each entire function numerically. It is also necessary to evaluate the functions over a common denominator so that only one complex division is performed. This is because the real part of the denominator of each term is -very close to zero in magnitude. When the computation 23 was not done in this order, ve found that the expected number in several systems which are supposed to achieve a steady state was of rapidly increasing negative magnitude. But when the same functions were computed in the suggested order, the expected numher in the systems approached steady state values nicely. Appendix III contains tahles of the numerical inversions of transient system descriptors for various different queueing systems. Recall from the introduction that those that reach steady state are characterized hy the ratio A/ky heing less than one. For these systems we note that the transient descriptors do indeed approach the steady state results. Moreover, this approach seems to he more rapid when A/ky is smallest and relatively slower for values of A/ky close to 1. This is as one would expect from the sensitivity analysis referred to in the introduction. Figures 2 through 6 are graphs of the data in tahles 7 through 9. These systems, for which A/ky = .67, are chosen "because they yield typical results for A/ky < 1. The initial conditions i = 0, 2, and h were chosen because 2 is the value of E n in the one-server case and close to E in the two-server case. Then the curves corresponding to i = show E (t) and E (t) increasing and E (t) decreasing while those corresponding to i = h show the opposite. When i = 2 a curious phenomenon is observed. Immediately after the system starts operating, E n (t) dips sharply and then climbs slowly back to the steady state value. This behavior characterizes all of the systems we investigated where the numher of units in the system initially and at steady state are nearly the same. We would next like to determine whether these systems we have studied are first order systems. When the system is empty initially, a first order system is one whose behavior can be described by the analytic function E n (t) = v(l- e - t/T ) where \) is the system's final value, and T is a time constant to be determined. It °° n After one time constant has elapsed, that is when t = T, a first order system will have attained 63.2% of its final value. After two time constants it will have reached 86.5% of final value and after five time constants, 99.3% of final value. For some of our systems which reach steady state, we approximate T "by one half the time required to achieve 86.5% of final value because this can he determined more accurately than the amount of time required to achieve 63.2% of final value. We find that when A/ky = .50, T = 2; when A/ky = . 6?,T = 3; when A/ky = .80, T = 20; and when A/ky = . 909,T = 125. In Figure 7 the percentage of final value attained is plotted against the number of time constants elapsed for the values of A/ky -t/T mentioned above when i = 0. We compare this to the function 1 - e ' , which is the portion of final value achieved at time t by a first order system. The curves intersect at t = 2T because of the definition used for T above. From Figure 7 we see that although the behavior of Poisson queueing systems with infinite queue length is similar to that of first order systems, they are not true first order systems. When t = T, the Poisson systems show values approximately 10% higher than would a first order system, while for t > 3T the Poisson systems are consist- ently 3% to 5% lower than a first order system. Had the time constant T been chosen differently these discrepancies would have been more pronounced. For systems which never reach steady state the functions E (t) and E (t) are growing steadily while E (t) is decaying, which is as one would expect when units arrive in the system faster than they can be serviced. The tables provide a quantified measure of this growth and decay. It is illustrated In Figures 8 and9» which show the number of units in the system and in the queue for the case I = and k = 1. Percent of Final Value Achieved 30 10Q% 90$ 10% 60% 50% ko% 30% 20% 10$ Time Constants Elapsed FIGURE 7 31 ftP -P •H a D |g(z)| on C, then f(z) and f(z) + g(z) have the same number of zeroes inside C [9]. We apply this to the denominator of (i.U). D is the disc |z| < 1, C is the circle |z| = 1, f(z) = (A+y+s)z and g(z) = Az 2 + y. On C, |f(z)| = |A+y+s| > |A+y| = |g(z)| because Re s > 0. f(z) has one zero inside C, at z = 0, so the denominator of (l.k) has one zero in |z| < 1. Now the zeroes of the denominator are a = * + y + s+ V(A+y+s) 2 - k\u 1 2A and !=&■ ? a = A+y+s- V(A+y+s) - hAy 2 2A Since |a 2 | < IclJ, a g must lie in |z| < 1. The convergence of n*(z,s) implies a, must also he a zero of the numerator of (I.J+). Then a 2 - y(l-a 2 )P (s) = and we know that i+l p o (s) = TH^p" ' (1.5) Substituting (20) into (19) gives z i+1 (l-z) i+l „* " (l-aj a 2 n (z 5 s) = —, — 2 . 11 -Mz-a )(z-a ) z i+1 (l-a 2 ) - (l-z)a^ 1 -A(z-a )(a-a )(l-a ) ' This can be factored as 38 i-1 (z-re ) \ z 1_,] a 2 - za 2 (z-a 2 ) i-l-j J z a" n # u..)- — &- ■i=o Xa i( 1 -^;) (z - a 2 )(1 - a 2 ) Cancelling (z-a ) from numerator and denominator gives i-1 z 1 -^ - a 2 y ^4 lT<*.«> -J=2. J^O. -l( 1 -^) (l -2 ) (l-a 2 ) \ z a 2 + a 2 ,1 = -C 1 - ^ Aol, z 1 "^ + |1 ^ ft 2 i+1 3 r 1 a 2 o , ; Xa-.(l-a ) (1.6) a I > 1 and |z| < 1 implies < 1 so 1 - z a. v-1 z a. £=0 39 Substituting this into (1.6) gives -£ Ift I_ ) \ /_z_\ ( ) \ i-J J ' '^0 \ V I [j=0 z a 2 i+1 + «2 IX /, Xo^d-ag) }^ o K)I (l - 7) By 1.2, P (s) is the coefficient of z injj (z,s). By expanding I. 7 in power: of z it can he determined that n>i j=0 a l J j=n+i+2 a l and P*f s )-I V W ,A\ n+1 V W. i p (t) n -(X+y)t n+2,j-i+l t ,j=i-n n-i+2j+l I n-i+2j+l j=n+i+2 *f.f I. . * where the argument of the Bessel function is 2/Ay t. Finally, we can use these two facts about Bessel functions 2v I (z) = I (z Z V v-1 W'> m i v («) = i. v («) ill to determine that p (t) = ~-(* + uH)/'/£ VI " Q / r-\i- n+ l I • + ( f\ I n-i WAV n+i+1 ♦ ft™ oo ■K> • s X j=n+i+2 *, (I. 10) for all n>0. APPENDIX II FINAL VALUES OF THE TRANSIENT SYSTEM DESCRIPTORS To determine the final values of the transient system descriptors E (t), E (t, and E (t), we employ the final value theorem for Laplace transforms, which states w» S f*(s) = ^; m f(t). [in s>0 t >0 ° When there is one server "we find lim _ /. v lim */ >. E = . E (t) = _ sE (s) n t-*>° n s->0 n i+1 lim |i . (X-y) ®2 s{ 1- — - — + - s>0 )s 2 s(l-a ) S d . (X-y)(l-a ) + sa lim 2 2 s-»0 s(l-a ) This is an indeterminate form of type — . We apply L' Hospital's Rule twice to find that this is equal to U-p) I- i^fl a(iti)4 ^f + s(i + r) £ U ^f lim \ 9s / \ 8a d a -2 -rf - s 1 85 8s 2 The first and second partial derivatives of a with respect to s are 9ot 2 _ 1 - ((ky+X+s) 2 - UkXy)"^ (ky+X+s) 3s " 2X *% ((ky+X+s) 2 - J4kXy)" 3/2 (ky+X+s) 2 - ((ky+X+s) 2 - UkXy) l/2 a 2 2A dS Letting s approach zero, we find U3 lim s->0 a 2 = 1 lim 2 _ 1_ s->0 8 s - (ky-X)" 1 (ku+A) 2A s 2 , . d a_ lim 2 s-K) r , 2 ds (ky-A)" 3 (ku+A) 2 - (ky-A)" 1 2A Substituting these values into the expression above gives s->0 n v ' y-A which is the well-known value of E stated in Chapter I. When there are two servers , lim *, , r, S E (s) s->0 n ; i + ■ lim A-2y , s (s+A+2y)a i+1 s-> s (sa 2 +Aa 2 +2y)(l-aJ 2ys6. o (l-a 2 ) 2 J sa +Aa +2y = i + lim / _ , » , , . s ^0 (2y-A)(a 2 -l)(sa 2 +Aa +2y) + (s +As+2y)aJ - 2ys6 . Q (l-a 2 f / (sa + Aa +2y ) (s- sa 2 ; ' This limit is an indeterminate form of type - q . We apply L' Hospital's Eule twice and find that it is kk 3%, I 3a 3a \ 3a (2 y+ ,a 2+ Aa 2 )(2 y -A) — | + 2 k, + b -gf + A-^y-X) — 3 a 3a 3 a Is 1 + 2 -g| + X 1 I (2y-X)(a 2 -l) 3s 3s • ^ 2 i-l/ 8a 2 X2 + (s 2 +sX+2ys)(i+l) a 2 | + (s +sX+2ys ) (i+l>iOg" I -g^ 3a . / \ I 3a 2 3 a g + 2(2s+A+2y)(i+l) ^ -^ + 2c* 2 ^ / J(2y+sa 2 +Xa 2 ) 1-2 — - s £ 3a 2 3a 2 W 3a g < 2|a 2+ s— + X— 1(1 -a 2 - s — 3 a 3 a 3 a g ., -— + S — 2 + X — ~J (S - Sa 2 } 3s 3s Taking the limit as s approaches 0, this "becomes Uy -X which is equivalent to the expression given for E n in Chapter I when k = 2. When there is one server we find that lim _ /. s lim */ s E = E (t) = ^ sE (s) m t->-°° m s-»-0 m , n i+1 t \ ,, (s+y)a Hm ± „ x + X=1L + 2 s->0 s p(l-a 2 ) W n ^2 v i+1 . . y(y-X)(a_-l) + (s +sy)a lim d g 1 ' 1 + s+0 " sy(l-a 2 ) h5 This is an indeterminate form of type — . We apply L'Hospital's Rule twice to find that it is 1 - -, , lim 1 / . \ 2 dS + 2(2s+y)(i+l)a i_^2 2 3s + 2 4 +1+ CB 2 *By)(i + l)^(4^[?/ 8 °< 2 8 2 a 2 - 2y — - sy — dS Taking the limit as s approaches this "becomes (x/ur l-U/y) which is the steady state result given hy (1.15) for E when k = 1 m In the two-server case we find _ lim _ / , s lim * , v E « . E (t ) = ^sE(s) m t-*» m s->0 m lim s->0 2 + ^i + (s+A+2y)a i+1 (l-a )(2y+s a 2 + A a 2 ) 2 " i 10 (l -2 ) 2y+Aa +sa i+1 i+1 2sa 2 '" 2s<5 -n 2 s(X+s)a .1. I,,, . ..I,,.,.. I | I I. ■ I , |. . . I. ■ . (l-a ) (2y+sa +Aa n ) 2y+Xa+sa y (l-a ) (2y+sa +Aa ) 2 2' 2 2' 2s6 iO 2y+Aa +sa [ k6 The limits of all terms involving 6 vanish and we are left with i - 2 + lim s-*0 y(2y-A)(a 2 -l)(2y+sa 2 +Aa 2 ) ,„ 2 , _ 2 2. 3n i+1 , , + (3s y+sAy+2sy +s A+s )a y( s-sa ) (2y+sa 2 +Aa 2 . This limit is an indeterminate form of type — , to which we apply L' Hospital's Rule twice. It then becomes i - 2 + lim s+0 8 a y(2y-A)(a 2 -l)l2 -~ 2 2 8 a 8 a + s - + A 8s 8s 8a / 8a 8a + 2y(2y _X)— a 2+ s— + A — 8 a + y(2y-A) - (2y+sa 2 +Aa 2 ) 8s 2 2 ! / i 2 + (3s y+sAy+2sy -s X+s )(i+l) — la — — 2 2 i da ? i+1 . + 2(6sy+Ay+2y +2sA+3s )(i+l) a —■ + (6y+2A+6s) a g ) / 8a 8 a 8 a \ / 8a g 3a 3a ? \ / 8a, \ + 5 — + A — ) + ^ Hit ■ ^ a 2 8 a. 8s ._., , (2y+sa 2 +Aa 2 ) hi Taking the limit as s approaches 0, this is evaluated as A 3 /y k/-x 2 which is equivalent to the steady state value E given "by (l.l6) when k = 2. Finally, we evaluate t, lim _ / , \ lim *, > E I = t- E I (t) = s-0 SE I (s) iim I ^2U + X +S )a^ +1 s6. s->0 Jy(l-a 2 )(2u+sa 2 +Xa 2 ) " 2y+sa +Aa The limit of the term involving 6. vanishes. We apply L' Hospital's Eule to the first term and take the limit as s approaches 0, giving 2 - A/y which is the steady state value for E when k = 2. APPENDIX III RESULTS OF THE NUMERICAL INVERSION OF THE LAPLACE TRANSFORMS This appendix contains the tabular data resulting from the numerical inver- sion described in Chapter IV. The following values of A/ky are used: .33, -50, .67 .80, .91, 1.0, 1.1. For X/ky < 1, the values of i are chosen so that the system can be observed approaching steady state in three ways. First we see how the system accumulates units when initially empty. Then we choose an initial condition close to steady state values. Third, we observe the number of units in system and queue decreasing to steady state values. Finally, when X/ky > 1 we see how systems which don't reach steady state grow. TABLE 1 49 A=l y=3 1= A/ky=.33 k=l y=1.5 k=2 Time E n (t) E m (t > E (t) n \^ E z (t 1.0 .38 .091 .65 .025 1.41 2.0 .1+6 .136 • 71 .054 1.38 3.0 .48 .15^ • 73 .070 1.36 4.0 .49 .161 .74 .077 1.35 5-0 .50 .164 .75 .080 1.35 6.0 • 50 .165 .75 .082 1.35 T.O .50 .166 .75 .083 1.34 8.0 • 50 .166 .75 .083 1.34 9.0 • 50 .166 • 75 .083 1.34 10.0 • 50 .167 • 75 .083 1.34 11.0 • 50 .167 • 75 .083 1.34 12.0 • 50 .167 • 75 .083 1.34 13.0 • 50 .167 • 75 .083 1.34 l4.0 • 50 .167 • 75 .083 1.33 15.0 • 50 .167 • 75 .083 1.33 16.0 • 50 .167 .75 .083 1.33 iT.o • 50 .167 • 75 .083 1.33 .18.0 • 50 .167 • 75 .083 1.33 19.0 .50 .167 • 75 .083 1.33 20.0 .50 .167 • 75 .083 1.33 25.0 .50 .167 • 75 .083 1.33 30.0 .50 .166 • 75 .083 1.33 4o.o • 50 .167 .75 .083 1.33 50.0 .50 .166 .75 .083 1.33 Steady State: • 50 .167 .75 .083 1.33 TABLE 2 .'*' A=l i= 1 A/ky = • 33 u=3 k=l y=l.5 k=2 Time E (t) n E m (t) E n (t) V*' E^t) 1.0 • 51 .165 .65 .061 1.29 2.0 .50 .163 .71 .076 1.33 3.0 .50 .162 • 73 .080 1.33 k.o .50 .165 .Ik .082 1.33 5.0 • 50 .167 .75 .083 1.33 6.0 .50 .166 • 75 .083 1.33 7.0 .50 .166 • 75 .083 1.33 3.0 • 50 .166 • 75 .083 1.33 9.0 • 50 .166 • 75 .083 1.33 10.0 • 50 .167 • 75 .083 1.33 11.0 • 50 .167 • 75 .083 1.33 12.0 • 50 .167 • 75 .083 1.33 13.0 • 50 .167 .75 .083 1.33 lU.O .50 .167 • 75 .083 1.33 15.0 • 50 .167 • 75 .083 1.33 16.0 .50 .167 • 75 .083 1.33 17-0 • 50 .167 • 75 .083 1.33 18.0 • 50 .167 .75 .083 1.33 19-0 • 50 .167 .75 .083 1.33 20.0 • 50 .167 .75 .083 1.33 25-0 .50 .167 • 75 .083 1.33 30.0 • 50 .167 • 75 .083 1.33 Uo.o .50 .167 .75 .083 1.33 50.0 • 50 .167 • 75 .083 1.33 Steady State: • 50 .167 • 75 .083 1.33 51 TABLE 3 X=l i =2 A/ky= .33 y=3 k=l y=l.5 k=2 Time E n (t) E (t) m E n (t) E (t) m E x (t) 1.0 .86 .37 .95 .11+8 1.08 2.0 .61 .2k .82 .115 1.26 3.0 • 55 .20 .76 .095 1.30 4.0 • 52 .18 • 76 .090 1.32 5-0 .51 .17 • 75 .087 1.33 6.0 .50 .17 • 75 .085 1.33 7.0 • 50 • 17 • 75 .083 1.33 8.0 .50 .17 .75 .083 1.33 9.0 • 50 • 17 .75 .083 1.33 10.0 • 50 .17 • 75 .083 1.33 11.0 • 50 • 17 • 75 .081+ 1.33 12.0 .50 .17 • 75 .081+ 1.33 13.0 • 50 .17 • 75 .081+ 1.33 ll+.O .50 .17 • 75 .083 1.33 15.0 • 50 .17 .75 .083 1.33 16.0 .50 .17 • 75 .083 1.33 17.0 • 50 .17 • 75 .083 1.33 18.0 • 50 .17 • 75 .083 1.33 19.0 .50 • 17 .75 .081+ 1.33 20.0 • 50 .17 • 75 .084 1.33 25.0 .50 .17 • 75 .081+ 1.33 30.0 • 50 .17 ■ 75 .083 1.33 1+0.0 • 50 • 17 .75 .083 1.33 50.0 • 50 .17 .75 .083 1.33 Steady State: • 50 .17 • 75 .083 1.33 TABLE 4 52 \=1 i= = A/k u =.5 y=2 k=l U=l k =2 Time E n (t) E (t) m E (t) n E (t) m E T (t) 1.0 .51 .Ik .84 .039 1.26 2.0 • 70 .26 1.02 • 11 1.17 3.0 • 79 .33 1.12 .17 1.11 4.0 .86 .38 1.18 .21 1.08 5-0 • 90 .1+1 1.22 .25 1.06 6.0 -92 .43 1.25 • 27 i.o4 7.0 .9k .45 1.27 .28 1.03 8.0 .96 .k6 1.29 • 30 1.02 9.0 .91 .47 1.30 • 30 1.02 10.0 • 97 .48 1.31 • 31 1.02 11.0 .98 .48 1.31 • 32 1.01 12.0 • 99 .49 1.32 • 32 1.01 13.0 • 99 .49 1.32 .32 1.01 lU.O .99 .49 1.32 .32 1.01 15.0 • 99 .49 1.32 .33 1.00 16.0 • 99 .49 1.33 .33 1.00 17.0 • 99 • 50 1.33 .33 1.00 18.0 1.00 • 50 1.33 • 33 1.00 19-0 1.00 • 50 1.33 .33 1.00 20.0 1.00 • 50 1.33 .33 1.00 25-0 1.00 • 50 1.33 .33 1.00 30.0 1.00 .50 1.33 • 33 1.00 4o.o 1.00 ■ 50 1.33 .33 1.00 50.0 1.00 • 50 1.33 .33 1.00 eady State : 1.00 • 50 1.33 .33 1.00 TABLE 5 53 X=l i =1 X/ky= .5 y=2 k-1 =1 k=2 Time E n (t) E (t) m E n (t) m E x (t) 1.0 .77 .29 .84 .11 1.07 2.0 .82 .35 1.02 .18 1.07 3.0 .87 .39 1.12 .22 1.05 4.o .90 .42 1.18 • 25 i.o4 5.0 • 93 .44 1.22 .27 1.03 6.0 • 95 .45 1.25 • 29 1.02 7.0 .96 .46 1.27 .30 '1.01 8.0 • 97 M 1.29 .31 1.01 9.0 • 98 .48 1.30 .31 1.01 10.0 .98 .48 1.31 .32 1.00 11.0 • 99 .49 1.31 • 32 1.00 12.0 • 99 .49 1.32 • 32 1.00 13.0 .99 .49 1.32 .33 1.00 14.0 .99 .49 1.32 .33 1.00 15.0 • 99 .49 1.32 .33 1.00 16.0 • 99 .50 1.33 • 33 1.00 17.0 1.00 • 50 1.33 .33 1.00 18.0 1.00 • 50 1.33 .33 1.00 19.0 1.00 • 50 1.33 .33 1.00 20.0 1.00 • 50 1.33 .33 1.00 25.0 1.00 .50 1.33 .33 1.00 30.0 1.00 • 50 1.33 .33 1.00 4o.o 1.00 • 50 1.33 .33 1.00 50.0 1.00 .50 1.33 .33 1.00 Steady State : 1.00 .50 1.33 .33 1.00 TABLE 6 54 \=1 i = 2 A/ky = • 5 y=2 k=l U=l k=2 Time E (t) n E (t) m E (t) n E (t) m E^t) 1.0 1.34 .67 1.31 .28 • 77 2.0 1.14 .58 1.28 .30 • 93 3.0 1.06 • 54 1.28 • 31 .97 4.0 1.03 • 52 1.28 ..31 • 99 5.0 1.01 • 51 1.29 .32 1.00 6.0 1.01 • 50 1.30 .32 1.00 7.0 1.00 • 50 1.30 .32 1.00 8.0 1.00 • 50 1.31 • 32 1.00 9.0 1.00 .50 1.32 .32 1.00 10.0 1.00 • 50 1.32 .33 1.00 11.0 1.00 .50 1.32 .33 1.00 12.0 1.00 • 50 1.33 .33 1.00 13.0 1.00 .50 1.33 .33 1.00 lU.o 1.00 • 50 1.33 .33 1.00 15.0 1.00 • 50 1.33 .33 1.00 16.0 1.00 • 50 1.33 .33 1.00 17.0 1.00 .50 1.33 • 33 1.00 18.0 1.00 • 50 1.33 .33 1.00 19.0 1.00 .50 1.33 .33 1.00 20.0 1.00 .50 1.33 .33 1.00 25-0 1.00 • 50 1.33 • 33 1.00 30.0 1.00 • 50 1.33 .33 1.00 40.0 1.00 • 50 1.33 .33 1.00 50.0 1.00 • 50 1.33 • 33 1.00 ; ady State: 1.00 • 50 1.33 .33 1.00 TABLE 7 55 A =2 i* =0 X/ky = .67 y=3 k=l y=1.5 k=2 Time E (t) n E m ^ E n (t) \^ EjCt) 1.0 .89 .37 1.25 .16 1.01 2.0 1.22 .63 1.59 .38 .86 3.0 1.1+1 • 79 1.79 • 5k • 79 4.0 1.51+ • 91 1.93 .65 .76 5-0 1.61+ 1.00 2.03 • 73 .7^ 6.0 1.71 1.06 2.10 • 79 • 72 7.0 1.76 1.11 2.15 .81+ .71 8.0 1.80 1.15 2.20 .88 • 70 9.0 1.83 1.18 2.23 • 91 .69 10.0 1.86 1.20 2.26 .91+ .69 11. 1.89 1.23 2.28 .96 .68 12.0 1.90 1.21+ 2.30 .98 .68 13.0 1.92 1.26 2.32 • 99 .68 ll+.O 1.93 1.27 2.33 1.00 .68 15.0 1.9k 1.28 2.31+ 1.01 .67 16.0 1.95 1.28 2.35 1.02 .67 17.0 1.95 1.29 2.35 1.02 .67 18.0 I.96 1.30 2.36 1.03 .67 19.0 1.97 1.30 2.37 1.01+ .67 20.0 1.97 1.31 2.37 1.01+ .67 25.0 1.98 1.32 2.38 1.05 .67 30.0 1.99 1.33 2.39 1.06 .67 1+0.0 2.00 1.33 2.39 1.06 .67 50.0 2.00 1.33 2.1+0 1.06 .67 Steady State: 2.00 1.33 2.1+0 1.07 .67 TABLE 8 56 A=2 i =2 A/ky=. y=3 k=l y=i.5 k=2 Time E (t) n E (t) m E (t) n E (t) m 1.0 1.59 • 90 1.61+ • 50 2.0 1.61 • 96 1.81 .62 3.0 1.67 1.02 1.93 • 70 k.O 1.72 1.07 2.03 • 77 5-0 1.77 1.12 2.10 .83 6.0 1.8l 1.15 2.15 • 87 7.0 1.8** 1.18 2.20 • 90 8.0 1.86 1.21 2.23 • 93 9.0 1.89 1.23 2.26 • 95 10.0 1.90 1.21+ 2.28 • 97 11.0 1.92 1.26 2.30 • 98 12.0 1.93 1.27 2.32 1.00 13.0 1.9k 1.28 2.33 1.01 ll+.O 1.95 1.28 2.3^ 1.01 15.0 1.96 1.29 2.35 1.02 16.0 1.96 1.30 2.35 1.03 17.0 1.97 1.30 2.36 1.03 18.0 1.97 1.31 2.37 1.04 19-0 1.98 1.31 2.37 l.Olf 20.0 1.98 1.31 2.38 1.05 25.0 1.99 1.32 2.38 1.05 30.0 1.99 1.33 2.39 1.06 Uo.o 1.99 1.33 2.39 1.06 50.0 2.00 1.33 2. UO 1.06 Steady State: 2.00 1.33 2. UO 1.07 E^t) • 71 .71+ •72 • 71 • 71 • 70 • 69 .68 .67 .67 .67 .67 .67 .67 .67 .67 .67 .67 .67 .67 .67 TABLE 9 57 A =2 i =1+ A/k U = • 67 y=3 k=l y=i.5 E m (t) k=2 Time E n (t) E (t) m E n (t) E z (t) 1.0 3.12 2.23 3.01+ 1.1+7 .29 2.0 2.61+ 1.87 2.72 I.29 .50 3.0 2.1+0 1.67 2.57 1.20 .58 k.O 2.26 1.56 2.1+9 1.11+ .62 5.0 2.18 1.1+9 2.1+5 1.11 .61+ 6.0 2.13 1.1+1+ 2.1+2 1.09 .65 7.0 2.09 1.1+1 2.1+1 1.08 .66 8.0 2.06 1.39 2.1+0 1.07 .66 9-0 2.05 1.38 2.39 1.07 .66 10.0 2.0*+ 1.37 2.39 1.06 .66 11.0 2.03 1.36 2.39 1.06 .66 12.0 2.02 1.35 2.39 1.06 • 61 13.0 2.01 1.35 2.39 1.06 .67 l*+.0 2.01 1.31+ 2.39 1.06 .67 15.0 2.01 1.31+ 2.39 1.06 .67 16.0 2.00 1.31+ 2.39 1.06 .67 17.0 2.00 1.31+ 2.39 1.06 .67 18.0 2.00 1.31+ 2.39 1.06 .67 19.0 2.00 1.31+ 2.39 1.06 .67 20.0 2.00 1.31+ 2.39 1.06 .67 25.0 2.00 1.33 2.1+0 1.06 .67 30.0 2.00 1.33 2.1+0 1.06 .67 1+0.0 2.00 1.33 2.1+0 1.06 .67 50.0 2.00 1.33 2.1+0 1.07 .67 ady State: 2.00 1.33 2.1+0 1.07 .67 TABLE 10 58 X=l 1= =0 x/ku*. 80 y=l.25 k=l y=.625 k=2 Time E (t) n m E n (t) E m ^ EjCt) 10 2.23 1.50 2.66 1.16 .55 20 2.85 2.08 3.28 1.73 .1+8 30 3.18 2.1+0 3.61 2.05 .k5 1+0 3.^0 2.62 3.8U 2.27 M 50 3.56 2.77 1+.00 2.1+1 .1+2 60 3.66 2.87 1+.10 2.51 .1+2 TO 3.73 2.9^ 1+.17 2.58 Al 80 3.78 2.99 1+.23 2.6k .kl \ 90 3.83 3.0J4 U.27 2.68 .1+1 100 3.87 3.07 14.31 2.72 .1+1 110 3.90 3.10 U.31+ 2.7 ] + .1+0 120 3.92 3.12 1+.36 2.76 .Uo 130 3.93 3.13 1+.37 2.78 .1+0 lUO 3.91+ 3.1U 1+.38 2.79 .1+0 150 3-95 3.15 1+.39 2.80 .1+0 160 3.96 3.16 1+.1+0 2.80 .Uo 170 3.97 3.17 kM 2.81 .1+0 180 3.97 3.17 k.k2 2.82 .1+0 190 3.98 3.18 1+.1+2 2.82 .1+0 200 3.98 3.18 kM 2.83 .1+0 250 3.99 3.19 1+.1+3 2.83 .1+0 300 1+ .00 3.20 k.kk 2.81+ .1+0 1*00 3.99 3.19 k.kk 2.Qk .1+0 500 1+.00 3.20 k.kk 2.8U .1+0 Steady State: 1+.00 3.20 k.kk 2.81+ .1+0 59 TABLE 11 X : .=4 \/k\i = .80 W=1.25 k=l u=.625 k=2 Time E n (t) m E n (t) E (t) m E z (t) 10 3.22 2.1+3 3.41 1.90 .44 20 3.36 2.58 3.67 2.13 .43 30 3.51 2.72 3.87 2.31 .42 4o 3.62 2.83 4.00 2.43 .42 50 3.70 2.91 4.11 2.53 .42 60 3.77 2.97 4.18 2.60 .41 70 3.81 3.02 4.24 2.65 .41 80 3.85 3.05 4.28 2.69 .40 90 3.88 3.08 4.31 2.72 .40 100 3.90 3.11 4.34 2.74 .40 110 •3.92 3.12 4.36 2.76 .40 120 3.9^ 3.14 4.38 2.78 .40 130 3.95 3.15 4.39 2.79 .40 l40 3.96 3.16 4.40 2.80 .40 150 3.96 3.16 4.40 2.81 .40 160 3.97 3.17 4.4l 2.81 .4o 170 3.97 3.17 4.42 2.82 .4o 180 3.98 3.18 1 4.42 2.82 .40 190 3.98 3.18 4.43 2.83 .40 200 3.99 3.19 4.43 2.83 .4o 250 3.99 3.19 4.43 2.84 .4o 300 4.00 3.20 4.44 2.84 .4o 400 3.99 3.19 4.44 2.84 .40 500 3.99 3.20 4.44 2.84 .40 Steady State: 4.00 3.20 4.44 2.84 .40 6o TABLE 12 X=l i= 8 x/ku= .80 u*1.25 k=l y=.625 k=2 Time E (t) n E (t) m E (t) n E .(t) m E T (t) 10 5-91 5.00 5.95 4.19 .20 20 4.99 4.l4 5.17 3.51 .32 30 4.57 3-75 U.83 3.21 .37 Uo 4.36 3-5U 4.67 3.06 .38 50 k.2k 3.43 U. 59 2.98 .39 60 4.l6 3.35 4.53 2.93 .39 TO i+.lO 3.30 4.50 2.90 .40 80 U.07 3.27 4.47 2.88 .40 90 U. 05 3.25 4.46 2.87 .40 100 k.ok 3.24 k.k6 2.86 .40 110 4.03 3.23 4.45 2.85 .40 120 4.02 3.22 U.U5 2.85 .40 130 It '.01 3.21 U.U5 2.85 .40 i4o U.01 3.21 4.44 2.8^4 .40 150 4.00 3.20 k.kk 2.81+ .40 160 4.00 3.20 k.kk 2.8)4 .40 170 4.00 3.20 k.kk 2.84 .40 180 U. 00 3.20 k.kk 2.84 .40 190 U .oo 3.20 k.kk 2. 84 .40 200 4.00 3.20 k.kk 2.84 .40 250 3.99 3.20 k.kh 2.84 .40 300 4.00 3.20 k.kk 2.84 .40 4oo 3-99 3-19 k.kh 2.84 .40 500 3.99 3.20 k.kk 2.84 .40 Steady State : 4.00 3.20 k.kh 2.84 .40 TABLE 13 6l X=l i= =0 A/ky= • 909 y=l.l k=l y=.55 k=2 Time E n (t) \^ E n (t) \^ E z (t) 20 3.8 2.9 1+.2 2.5 .31+ 1+0 5-1 1+.2 5.5 3.8 .28 6o 5-9 5.0 6.3 1+.6 .26 80 6.5 5.6 7-0 5.2 .23 100 T.o 6.1 l.k 5-7 .22 120 7.3 6.5 7-8 6.0 .21 ll+Q 7.6 6.8 8.1 6.3 .21 160 7.9 7-0 8.4 6.6 .21 180 8.1 7-2 8.6 6.8 .20 200 8.3 l.k 8.8 7.0 .20 220 8.5 1.6 9.0 7-2 .20 2^0 8.6 7-7 9-1 7-3 .19 260 8.8 7-9 9.2 7-1+ • 19 280 8.9 8.0 9.1+ 7-5 .19 300 9-0 8.1 9.1+ 7-6 • 19 320 9.1 8.2 9-5 7-7 • 19 3I+O 9.1 8.2 9.6 7.8 • 19 360 9-2 8.3 9-7 7-9 .19 380 9-3 8.k 9.8 8.0 • 19 Uoo 9.h 8.4 9.8 8.0 .19 500 9.6 8.7 10.0 8.2 •19 600 9-7 8.8 10.2 8.1+ .18 800 9.8 8.9 10.3 8.5 .18 1000 9-9 9.0 10.1+ 8.6 .18 Steady State: 10.0 9.1 10.5 8.7 .18 TABLE Ik \=1 i=10 A/ky= • 909 u-1.1 k=l y-55 k=2 Time E (t) n E (t) m E (t) n E (t) m. E I (t) 20 8.5 7.6 8.6 6.7 .13 ko 8.2 7-3 8.3 6.6 .18 60 8.2 7-3 8.U 6.6 .19 80 8.3 l.h 8.6 6.8 .20 100 Q.k 7-5 8.7 6.9 .20 120 8.5 7.6 8.9 7-1 .20 lUo 8.6 7.7 9.0 7.2 .19 160 8.8 7-9 9-1 7-3 .19 18 8.9 8.0 9-3 7.5 .19 200 9-0 8.1 9.U 7.6 .19 220 9-1 8.1 9.5 7-7 .19 2U0 9.1 8.2 9-6 7-7 .19 260 9.2 8.3 9.6 7.8 .19 280 9-3 8.U 9-7 7-9 .19 300 9-3 8.1+ 9-8 7-9 .19 320 9.h 8.5 9-8 8.0 .19 3U0 9.h 8.5 9-9 8.1 .19 360 9-5 8.6 9-9 8.1 .19 330 9-5 8.6 10.0 8.2 • 19 i+00 9.6 8.6 10.0 8.2 .19 500 9.7 8.8 10.1 8.3 .18 600 9.8 8.9 10.3 8.U .18 800 9-9 9-0 10.3 8.5 .18 1000 9.9 9.0 10.il 8.6 .18 Steady State : 10.0 9-1 10.5 8.7 .18 63 TABLE 1 15 X=l i=20 A/ky= .909 u=i.i k=l y=5-5 k=2 Time E (t) n E (t) m E (t) n E- (t) m E T (t) 20 18.0 17.0 18.0 16.0 .006 1+0 16.2 15.3 16.2 11+.3 .0)43 6o Ik. 9 13-9 11+.9 13-0 .081 8o 13.9 12.9 ll+.O 12.1 .11 100 13-1 12.2 13-3 11.1+ .13 120 12.6 11.6 12.8 10.9 .Ik 1U0 12.1 11.2 12.1+ 10.5 .15 160 11.8 10.9 12.0 10.2 .16 180 11.5 10.6 11.8 10.0 .16 200 11.3 10.1+ 11.6 9-8 .17 220 11.1 10.2 11.1+ 9-6 .17 2l+0 11.0 10.0 11.3 9.5 • 17 260 10.8 9-9 11.2 9-k .17 280 10.7 9-8 11.1 9.3 .17 300 10.6 9.7 11.0 9.2 .17 320 10.5 9.6 10.9 9-1 .18 3I+O 10.5 9-6 10.9 9-1 .18 360 10.1+ 9-5 10.8 9.0 .18 380 10.1+ 9-5 10.8 9.0 .18 1+00 10.3 9.1+ 10.8 8.9 .18 500 10.2 9-3 10.6 8.8 .18 600 10.1 9.2 10.6 8.7 .18 800 10.0 9-1 10.5 8.7 .18 1000 10.0 9-1 10.5 8.7 .18 Steady State : 10.0 9-1 10.5 8.7 .18 6k TABLE 16 X=l i= X/ky=l M=l k=l u=.5 k= 2 Time E n (t) m B a Ct) E (t) m B x (t) 1.0 • 70 .23 1.12 .066 1.09 2.0 1.15 .53 1.56 .21+ • 87 3.0 1.1+8 .80 1.90 .1+5 .72 k.O 1.79 1.06 2.21 .68 .63 5-0 2.06 1.31 2.50 .90 .56 6.0 2.29 1.52 2.73 1.10 .50 7.0 2.50 1.71 2.95 1.28 .1+6 8.0 2.70 1.90 3.15 1.1+6 .1+3 9.0 2.89 2.08 3.35 1.61+ .1+0 10.0 3.08 2.26 3.5^ 1.82 .38 11.0 3.26 2.1+3 3.72 1.98 .36 12.0 3.1+3 2.59 3.89 2.1I+ .31+ 13.0 3.58 2.7^ k.ok 2.28 .32 lU.O 3.72 2.88 1+.19 2.1+2 • 31 15.0 3.87 3.01 1+.33 2.56 • 30 16.0 i+.oi 3.15 1+.1+7 2.69 .29 17.0 U.15 3.29 1+.62 2.83 .28 18.0 U .29 3.1+3 It. 76 2.96 .27 19.0 I*. 1+3 3.56 1+.90 3.09 .26 20.0 )4.56 3.68 5-03 3.22 .26 25.0 5.13 1+.21+ 5.60 3.78 .23 30.0 5.68 14.79 6.16 1+.31 .21 1+0.0 6.6l 5.70 7-09 5.23 .18 50.0 7.1+6 6. 51+ 7- 91+ 6.06 .16 No Steady State Solution 65 TABLE 17 X=l i= =0 X/ky=l.l u=-9 k=l y=.^5 k=2 Time E (t) n m E (t) n E (t) m E I (t) 1.0 .73 ,2k 1.15 .070 1.07 2.0 1.21 • 57 1.63 .26 .81+ 3-0 1.57 .87 2.01 .1+9 .68 k.o 1.93 1.18 2.37 • 76 .58 5.0 2.25 l.i+7 2.70 1.02 • 50 6.0 2.53 1.73 2.98 1.26 .1+1+ 7.0 2.78 1.96 3.2U 1.1+8 .1+0 8-0 3.02 2.19 3.U8 1.7-1 .36 9.0 3.27 2.1+2 3-73 1.93 .33 10.0 3.51 2.66 3.98 2.16 .31 11.0 3.75 2.88 U.21 2.38 .29 12.0 3.96 3.09 k.k3 2.58 .27 13.0 1+.16 3.28 h.63 2.77 .25 ll+.O 1+.35 3-U6 1+.82 2.96 .21+ 15-0 I+.5I+ 3.65 5.02 3.1U .23 16.0 k.lk 3.81+ 5.21 3.33 .22 17.0 1+.93 1+.03 5.U1 3.52 .21 18.0 5-13 1+.22 5.61 3.71 .20 19.0 5.32 k.kl 5.80 3.90 • 19 20.0 5.51 1+.59 5.98 I+.07 .18 25.0 6.33 5Al 6.81 1+.88 • 15 30.0 7-17 6.23 7.65 5.70 .13 1+0.0 8.63 7.69 9.12 7.15 .10 50.0 10.10 9.10 10. 5^ 8.56 .081+ No Steady State Solution 66 TABLE 18 X=l i= 2 X/ku=l.l u=.9 k=l u=.45 k=2 Time E (t) n E C.t) m E (t) n E (t) m E I (t) 1.0 2.1k 1.25 .58 • 57 .36 2.0 2.36 1.51 • 75 .88 .40 3.0 2.58 1.74 .94 l.ll .39 1+.0 2.83 1.99 1.18 1.38 .37 5.0 3.09 2.24 1.43 1.63 .34 6.0 3.31 2.45 1.64 1.85 .32 T.o 3.52 2.65 1.84 2.05 .30 8.0 3.73 2.86 2.05 2.25 .28 9.0 3.94 3.07 2.26 2.46 .26 10.0 k.l6 3.28 2.48 2.68 • 25 11.0 4.38 3.49 2.69 2.88 .24 12.0 4.57 3.68 2.89 3.07 .23 13.0 4.75 3.86 3.07 3.25 .22 lU.O 4.93 4.03 3.25 3.43 .21 15.0 5.11 4.20 3.43 3.60 .20 16.0 5-29 4.38 3.61 3.78 .19 17.0 5.48 4.56 3.80 3.96 .18 18.0 5.66 4.75 3.98 4.15 .17 19.0 5.85 4.93 4.17 4.33 .17 20.0 6.02 5.10 4.34 4.50 .16 25.0 6.81 5.88 5.14 5.28 .14 30.0 7.62 6.68 5.96 6.08 .12 Uo.o 9.05 8.10 7.4l 7-51 .10 50.0 10.45 9.49 8.82 8.90 .081 No Steady State Solution 67 LIST OF REFERENCES [lj Bailey, N. T. J., "A Continuous Time Treatment of a Simple Queue Using Generating Functions," Journal of the Royal Statistical Society - Series B , Vol. 16, No. 2, 195h, pp. 288-291. [2] Beightler, C. S., L. G. Mitten and G. L. Nemhauser , "A Short Table of z- Transforms and Generating Functions," Operations Research , Vol. 9, No. h, 1961, pp. 57^-577 - [3] Bowdon, E. K. , Sr . , "Modeling and Analysis of a Network of Computers," Ph.D. dissertation, University of Iowa, Iowa City, Iowa, 1969- [h] Cox, D. R. , and W. L. Smith, Queues , Methuen and Company, London, 1961. [5] Farber , D. J., and K. C. Larson, "The System Architecture of the Distributed Computer System - An Informal Description," University of California, Irvine, California, Technical Report No. 11, September, 1971, 36 pages • [6] Feller, ¥. , An Introduction to Probability Theory and Its Applications , Vol. I , 3rd Ed., John Wiley and Sons, New York, 1968. [7] Kaplan, ¥. , Advanced Calculus , Addison - Wesley, 1952. [8] Kendall, D. G., "Some Problems in the Theory of Queues," Journal of the Royal Statistical Society - Series B , Vol. 13, No. 1, 1951, pp. 151-173- [9] Levinson, N. , and R. M. Redheffer, Complex Variables , Holden - Day, San Francisco, 1970. [10] Lindley, D. V., "The Theory of Queues with a Single Server," Proceedings of the Cambridge Philosophical Society , Vol. kQ , No. 2, 1952, pp. 277-289- [11] Magnus, W. , and F. Oberhettinger , Formulas and Theorems for the F unctions of Mathematical Physics , Chelsea, New York, 195^- [12] Saaty, T. L., Elements of Queue ing Theory , McGraw - Hill, New York, 1961. 68 [13] Saaty, T. L. , "Time - Dependent Solution of the Many - Server Poisson Queue," Operations Research Vol. 8, No. 6, i960 , pp. 755-772. [lU] Wagner, H. H. , Principles of Operations Researc h, Prentice - Hall, Englewood Cliffs, New Jersey, 1969. [15] Weeks, W. T. , "Numerical Inversion of Laplace Transforms Using Laguerre Function," Journal of the Association for Computing Machinery , Vol. 13, No. 3, 1966, pp. 1*19-1*29. BLIOGRAPHIC DATA IEET 1. Report No. UIUCDCS-R -73-579 _L 3. Recipient's Accession No. Tit [e and Subtitle Time -Dependent Descriptors for the Poisson Queue 5. Report Date June 1973 6. Author(s) Rhoda Hornkohl Barr 8. Performing Organization Rept. No - uiucdcs-r -7 3 -57 9 Performing Organization Name and Address University of Illinois at Urbana-Champaign Department of Computer Science Urbana, Illinois 6l801 10. Project/Task/Work Unit No. II. Contract /Grant No. NSF GJ 28289 Sponsoring Organization Name and Address National Science Foundation 1800 G Street, N.W. Washington, D.C 20550 13. Type of Report & Period Covered Master of Science The sis 14. Supplementary Notes Abstracts he theory of queueing systems at steady state is more advanced than that of systems t arbitrary points in time for two reasons. First, many systems are assumed to pproximate steady state after a relatively short period of time . Second, steady state •nalysis is easier to do than time -dependent analysis. While steady state analysis rovides much useful information about queueing systems, it gives no indication of .ow long it takes a system to reach steady state or how the system behaves before eaching it. Further, many systems never reach steady state, so steady state analysis annot be applied at all in these cases. In this paper we define time -dependent system escriptors which are analogous to some well-known steady state descriptors for a bisson system with infinite queue length. We discuss some transient solutions for hese systems and use them to determine the Laplace transforms of the transient .escriptors we have defined. Finally, we determine the transient descriptors explicit!^ y inverting their Laplace t ransforms numerically. Key Words and Document Analysis. 17a. Descriptors Identifiers/Open-Ended Terms • COSAT1 Field/Group Availability Statement Release Unlimited IM NTIS-35 ( 10-70) 19. Security Class (This Report) UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21. No. of Pages 71 22. Price S USCOMM-DC 40329P7 1 AUG 1 1 1973 \ tf % ■" ■■"''""''"" l lljniMiUHJB I IIinilffllMWWr'-ftW 'tfflW UNIVERSITY OF ILLINOIS-URBANA 510.84 IL6R no. C002 no.577-582(1973 On-line filing (OLF) a program package 3 0112 088400707