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GEOMETRIC PROGRAMMING AND CONDENSATION

BY

KIMBERLY OSTER

ARTHUR LAOS LIIMAIO COLOSMiDO SCfiOOL ci MINES

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All rights reserved INFORMATION TO ALL USERS

The qu ality of this repro d u ctio n is d e p e n d e n t upon the q u ality of the copy subm itted. In the unlikely e v e n t that the a u th o r did not send a c o m p le te m anuscript and there are missing pages, these will be note d . Also, if m aterial had to be rem oved,

a n o te will in d ica te the deletion.

uest

ProQuest 10783661

Published by ProQuest LLC(2018). C op yrig ht of the Dissertation is held by the Author. All rights reserved.

This work is protected against unauthorized copying under Title 17, United States C o d e M icroform Edition © ProQuest LLC.

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A thesis subm itted to the Faculty and the Board of T rustees of the Colorado School of Mines in partial fulfillment of the requirem ents for the degree of M aster of Science (Mathematics).

Golden, Colorado Date: II j*>o Signed: ^ ( ■ C ... Kimberly M. O ster Approved: Thesis Advisor Golden, Colorado Date: l j \ Z q \ 9 o — -Dr. Ardel J . Boes Professor and Head D epartm ent of M athem atics

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ABSTRACT

Many industrial queuing problem s can be modelled as M /G /1 queues or M /M /s queues w ith parallel servers. With given param eters, the optim al service capacity can be determ ined through linear search, as the cost functions are unim odal.

This stu d y p resen ts a proof, assum ing the single server is capable of providing service a t the sam e rate as all s servers combined, th a t M /M /s queues are economically optim al only if s = 1. T hat is, it is economically preferable to build a single facility with a large capacity th a n m ultiple servers w ith the sam e total capacity.

The thesis th e n develops a convergent algorithm capable of solving economic queuing problem s where th e queue h a s unlim ited queue length, unlim ited calling population, first-in first-out service order and is either M /M /1 or M /G /l . The algorithm u se s condensation m ethods to reduce the one and three degree of difficulty, constrained, signomial problem s to zero degree of difficulty, posynom ial problem s which are th en directly solvable by conventional geometric program m ing techniques.

The algorithm represents a m ore direct m ethod for estim ation of the optim um th a n the linear search, an d in m ost problems, requires less total calculations. A variety of hypothetical problem s were used to successfully

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te st the algorithm and th e resu lts were verified by linear search.

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TABLE OF CONTENTS

ABSTRACT... iii

LIST OF FIGURES AND TA B LES... vi

ACKNOWLEDGEMENTS ... vii

C hapter 1: AN INTRODUCTION TO INDUSTRIAL QUEUING M ODELS... 1

C hapter 2: DERIVATIONS FROM QUEUING T H E O R Y ... 15

Single Server Q ueues ... 15

A M ulti-server Q ueue ... 22

C hapter 3: GEOMETRIC PROGRAMMING AND THE SINGLE SERVER Q U E U E ... 30

Geometric Programm ing Overview . ... 31

Distorting GP to Solve the Single Server Q ueue . ... 34

C hapter 4: SAMPLE RUNS OF THE ALGORITHM AND EXPLORATION OF THE QUEUE M O D E L ... 40

The M /M /1 E x a m p le s ... 42

The M /G /1 E x a m p le s ... 48

C hapter 5: CONCLUDING STATEMENTS AND RECOMMENDATIONS FOR FURTHER RESEARCH ... 54

REFERENCES C IT E D ... 57

APPENDIX A: FLOW CHARTS OF GEOMETRIC PROGRAMMING CONDENSATION ALGORITHMS... 60

APPENDIX B: SPR EA D SHEET VERIFICATION OF GP CONDENSATION ALGORITHM R ESU LTS... 63

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LIST OF FIGURES AND TABLES

Figure 1: O peration Costs versus Service L evel:... 4

Table 4.1: Example Problems Used for A n a ly s is ... 41

Table 4.2: R esults of Problem M /M /1 A ... 43

Table 4.3: R esults of Problem M /M / I B ... 44

Table 4.4: R esults of Problem M /M /1 C ... 45

Table 4.5: R esults of Problem M /M /1 D ... 46

Table 4.6: R esults of Problem M /M / I E ... 47

Table 4.7: R esults of Problem M / G / 1 A ... 49

Table 4.8: R esults of Problem M / G / I B ... 49

Table 4.9: R esults of Problem M / G / 1 C ... 50

Table 4.10: Results of Problem M / G / I D ... 52

Table 4.11: R esults of Problem M /G / I E ... 52

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ACKNOWLEDGEMENTS

I would like to take th is opportunity to th a n k the m em bers of my thesis committee, Dr. R uth M aurer and Prof. William Astle, for the assistan ce and advice, and to especially th a n k Dr. R. E. D. Woolsey for is guidance and assistance, n ot only in the evolution of th is thesis, b u t th ro u g h o u t my post-graduate p u rsu its.

I would also like to th a n k my family for the patience and understanding. I owe m u ch to m y friends here a t m ines for helping me to m aintain a realistic perspective, foremost J a n Caffey, Jo e Katz, and T am ar Raphaeli, as well as my friends elsewhere.

Finally, I’d like to generally th a n k the guild; the team w ork and cam araderie is truly responsible for m y achievements.

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Chapter 1

AN INTRODUCTION TO INDUSTRIAL QUEUING MODELS

Consider a hypothetical shipping company, or division of a

company. Among other m ethods of transport, th e com pany u se s cargo ships between several ports. This necessitates the com pany’s

ownership, or a t least use, of docking and loading/unloading facilities a t each port it utilizes. The related costs to the com pany of th is docking operation include th e capital costs of the docking equipm ent, salaries paid to the employees an d other various m aintenance costs.

Additionally, if the efficiency of the facility is su ch th a t a ship m u st w ait outside the dock while a previous ship is being loaded or unloaded, the m aintenance costs of th a t waiting ship are also p a rt of th e dock’s cost of operation. The com pany w ants to know how m any docks it should

operate a t each port, and a t w hat service level to operate each. This system , and system s like it, where arriving traffic

accum ulates in a line and dem ands service, are known as queues, or waiting lines. A queue is defined by six characteristics. The interarrival tim es are random variables from a given interarrival distribution. The

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service tim es are random variables from a given service distribution. There are a specified num ber o f servers, which have a clearly defined order o f service (i.e. first-in first-out (FIFO), last-in first-out (LIFO), prioritized, etc.). Finally, declaring the size o f th e p o ten tia l callin g population and the m axim um queue len gth fully defines the queue. For th is shipping example, only random arrivals (i.e. those whose interarrival tim es are independent, identically distributed, random variables from the m em oiy-less exponential distribution), from a potentially infinite calling population, a t a FIFO queue of unlim ited length, are relevant.

The accepted notation for queues is the Kendall-Lee form at (Winston 1987) w hich lists the six characteristics, in the order given above, separated by vertical slashes. Exponential interarrival time distributions are indicated by an M, as are exponential service time distributions. O ther distributions of interest are a general distribution (G or GI), the Erlang (EJ, and determ inistic (D) behavior. It is a common practice to list only the first three characteristics w hen dealing with a FIFO queue of infinite length and infinite calling population. Using this notation, M /G /1 and M /M /s queues are those relevant to th is problem.

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responsible for both the servers (in th e above case, the docks) and the custom ers (the ships). The total expected cost of queue operation can be split one of two ways. Van Voorhis (1956) divided it into the cost due to custom ers waiting for servers (waiting time) and th a t due to servers waiting for custom ers (idle time). We assum e th a t both these costs are linear functions of tim e (i.e. waiting cost is linear function of waiting time and idle cost is a linear function of idle time). Since the waiting time is inversely proportional to the service capacity (the nu m b er of servers multiplied by their service rate) while the idle time is directly

proportional to the service capacity, the waiting and service costs are also inversely and directly proportional to service level, respectively. If the total cost is the su m of the two, the m inim um total cost is a t the intersection of these two functions where the waiting cost equals th e idle cost (see fig. 1) and any change in th is total cost is instigated by a

change in behavior of either the servers, the custom ers, or both.

Hillier (1964), on the other hand, split the total cost between the sam e waiting cost and the cost of servicing custom ers. The service cost is assum ed to be directly proportional to service rate. Thus, the service cost, like the idle cost, is directly proportional to th e service capacity. The total cost rem ains the su m of the service and waiting costs. The

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— Idl e Cost — Waiting Cost •^|£- Total Cost

Figure 1: Operation Costs versus Service Level

Source: Van Voorhis, W. R 1956. Waiting line theory as a m anagem ent tool. O perations Research 4: 233.

m inim um of th e total cost rem ains a t the intersection of the two functions where the waiting cost is equal to the service cost.

In his work in 1963, Hillier proposed several cost models for m inim ization of total expected cost (TEC); each is suited to the different decision variables common to su ch a model. The first is intended for a situation where the nu m b er of servers is the only unknow n. The second adds the m ean arrival rate to the list of unknow ns. The third (and final) model is designed for the case w hen both the num ber of servers and the

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m ean service rate are undeterm ined1. This is the appropriate model for th e shipping problem introduced above, where the individual docks are the servers and the service rate is dependent on the num ber of workers and the equipm ent employed.

Hillier defined the expected service cost as C , x s x f ( n ) ,

w here C8 is the cost of service per server per custom er ($/server/custom er), s is the num ber of servers, p is the m ean service rate (custom er/tim e), and f is the ’’ratio of the m arginal cost of service a t (service rate) p to the m arginal cost of service w hen the average service time is one u n it of time."2 Thus, the expected service cost h a s u n its of $ /tim e.

The expected waiting cost (also with u n its of $/tim e) is X t ™ X Ljfr ^ X

where is the cost of tim e per custom er per u n it time ($/custom er/tim e), and L, the expected num ber of custom ers in the queue system a t a given time is equal to the sum of L*. the expected n um b er of custom ers in the waiting line, and Ls, the expected nu m b er of

^ o t e the second an d third models are reversed in Hillier and Lieberman (1986).

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custom ers In service.

The total expected cost is the sum of the expected service and waiting costs. The goal is to minimize the TEC (i.e. minimize th e long ru n expected total cost per u n it time). The fact th a t TEC is a long ru n expected value indicates th a t the system m u st reach a steady (or

equilibrium) state. For th a t to occur in any queue the m ean service rate (p) m u st be greater th a n the m ean arrival rate (A.). If p < >», then, in the long ru n , the waiting line becom es infinitely long and can never be emptied. W hen X < p, the queue length varies and expected behavior is relatively easy to derive given an exponential interarrival time

distribution.

Note th a t total expected cost is not the only m easure of system performance. There are others su c h as the average w ait of all the custom ers, the average w ait of the custom ers who m u st wait, the

percent of the custom ers who w ait longer th a n a given acceptable time, and the total am ount of idle server time. Shelton (1960) discusses these and other m ethods more thoroughly.

Historically, two approaches have been tak en to th is type of problem: sim ulated sam pling (or simulation); and optimization through m athem atical modeling. Sim ulation answ ers the question of w hat

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o u tp u t resu lts from given input. Optimization does the opposite, determ ining w hat in p u ts are necessary to cause given o u tp u t values.

Optimization h a s itself been divided into two areas: control and design. Control involves determ ining an optimal operation policy of a given queue. T hat is, minimizing the costs of operation of an existing queue by tu rn in g servers off and on and vaiying their service rates as a response to the s ta tu s of the queue; it is a m atter of dynam ic decision models. Designing, static decision modelling, focuses on determ ining an optim al queue assum ing th a t all potential will always be used. In other words, all servers p resen t will be tu rn ed on and functioning a t the m axim um possible service rates; it is assum ed there is no advantage to dynam ically vaiying the queue’s serving capacity.

All three of these queue-related topics (simulation, control, and design) have been subjects of considerable research since th e late

1950’s.

First, it is helpful to consider a few examples of work in queue sim ulation. Van Voorhis (1956) provided a basic example in his

sim ulation of a full service gas station with random service and arrivals. Setting the salary of station a tte n d a n ts and the average revenue per custom er, while allowing th a t a custom er arriving a t a long queue m ay

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choose to leave, the au th o r ra n th e model for various n um bers of

a tten d a n ts an d determ ined the b est m anning policy. This was a simple example, more easily solved by m athem atical modelling and optimization b u t m any queues are so intricate th a t direct m athem atical modelling is extremely difficult.

The w arehouse docking model of Schiller and Lavin (1956) is a more involved example. Wiebolt Stores, Inc. of Chicago w as operating three w arehouses b u t planned to consolidate to a single w arehouse in the n ear future. There were constraints on the access an d servicing ability of the single w arehouse which where com plicated by varying m ean arrival rates a t different tim es of the day. By ru n n in g the model u n d er different scenarios the au th o rs determ ined the optim al num ber of new docks to add to the w arehouse.

More recently, Chelst, Tilles, and Pipis (1981) used sim ulation in a coal tran sp o rt problem. The com pany in question used a single

unloader to service incoming coal cars. C onstant u se caused frequent down-time for m aintenance, and coal cars waiting in cold w eather often arrived with frozen coal th a t dam aged the facility during processing. Using sim ulation, the au th o rs evaluated the addition of a second unloader and the possibility of alternating operation between the two.

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Control design emerged in the early 1960’s w hen Yadin and Naor (1963) were the first in a series of m athem aticians to investigate the optimal operation policy of the M /G / 1 queue using m athem atical

modelling and optimization. The a u th o rs recognized th a t for a queue to attain a steady state there m u st be excess service capacity and therefore idle periods for the server. Their algorithm was based upon reducing the idle fraction of the server’s tim e by sh u ttin g the server off w hen no

custom ers are present. Due to set-up and shut-dow n costs it is not optim al to tu rn the server back on immediately w hen a new custom er arrives. Yadin and Naor derived a value, R, dependent on set-up, s h u t­ down, waiting, and idle costs as well as the six basic queue

characteristics, su ch th a t w hen the nu m b er of waiting custom ers reaches or su rp asses R, the server is returned to operating sta tu s.

In 1967, the sam e au th o rs addressed control from the angle of varying the service rate an d outlined an optimization procedure su ch th a t the service rate w as a function of the num ber of custom ers waiting as well as th e recent history of the system.

H eym ann (1968) furth er investigated Yadin an d Naor’s M /G /l w ith removable server. D iscounting the costs due to the long term n atu re of the problem, he obtained an equation for the expected

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discounted cost as a function of the sam e value R and the interest rate. Heym ann also presented a recursion relation to find the optimal

dynam ic policy for undiscounted problems.

Later th a t sam e year, H eym ann and M arshall extended this field of dynam ic control by server removal to the G I/G /1 queue. Considering undiscounted problem s, they bounded the cost rate and the optimal policy.

In 1971, Bell provided the optim ality proof for Heym ann’s discounted M /G /1 model, as well as an "improved com putational algorithm" for determ ining R and the optimal policy.

The following year, Crabhill revived the stu d y of queue control through variance of service rate. A ssum ing k possible service rates and a cost stru c tu re su ch th a t one cost is dependent upon the service rate and another on the nu m b er of custom ers waiting, Crabhill presented an algorithm to derive optim al policy and dem onstrated it for the case k is two.

Most recently, Bell (1980) extended his research to the M /M /2 undiscounted queue an d Szarkowicz (1985) analyzed both discounted and undiscounted M /M /s queues with removable servers.

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work w as done by Mangelsdorf (1955), who introduced the u se of economic models to analyze waiting line problems. He used the

convention of idle and waiting costs (versus service an d waiting costs) an d calculated optimal ratios of these costs for given, fully defined, queues. M angelsdorf s focus w as m ainly on m achine assignm ent problem s which are subject to a limited calling population.

In 1963, Hillier entered the field by introducing his three queue cost models m entioned above. All of his models assum e a n infinite calling population, in co n trast to Mangelsdorf, b u t the two au th o rs agreed th a t one of the more difficult aspects of queue design is

determ ining the waiting cost. Hillier drew a com parison between waiting cost and the stock o ut costs of inventory, hoping to simplify the problem.

Hillier (1964) describes the general approach appropriate for queuing models and sum m arizes the stru ctu re of the model. He also m entions a problem, sim ilar to the shipping problem addressed here, involving a crew, performing together as a single server, whose service rate can be altered by addition of more crew m em bers, equipm ent upgrade, etc.

In 1967, Hillier and U eberm an collaborated on a text which includes a sum m ary of Hillier’s 1964 paper. Additionally, they discuss

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two cases in the form ulation of waiting cost functions. The first case covers custom ers who are external to the service organization. In this situation the waiting cost is due to loss of future b u sin ess and is

proportional to the w ait experienced by each individual custom er. The second case is for internal custom ers whose waiting cost is due to lost productivity and is, therefore, proportional to the n u m b er of custom ers stu ck in the queue. The au th o rs also review the effect on cost of the custom ers’ travel time to an d from the server.

Stidham (1968) (as well as his article of 1970) u ses Hillier’s models to attack general queues. Using stochastic m inim ization he proves

single server optim ality for the G /M /s, G /D /s , and G /E ^ /s queues. He later extends th is resu lt to service costs th a t are non-linear b u t concave and waiting costs th a t are monotone increasing in the waiting time. He also addresses queue netw orks and non-FIFO queues.

Most of the above research (with the exception of Stidham) h a s m ade the assum ption of service costs th a t are directly proportional to the service rate. T hat is, f(p)=p, and the service cost is sC8p. This is not always an accurate assum ption and, in the case of the shipping

problem, m ay not be a t all correct.

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to the theory and u se s of the learning curve. The theory behind learning is simply th a t the more frequently an act is performed the more efficient the perform er becomes; the perform er learns as the work is done. The goal of learning theory is to uncover an efficiency im provem ent rate regular enough to be predictable.

Andress introduces three learning curve form ulas, the la st of which describes learning in a queue. The form ula is as follows:

T = kxn* \

for T, the total m an h o u rs required to build x units, k, the cost in m an h o u rs of the first unit, and n, the learning index.3 This effect appears in the service cost of the shipping problem where f(p) can now be defined so th a t

Total Service Cost

-S ubstituting m = n + 1, fl(p) becomes the following:

/ ( H ) - , | i » ( r o s l ) .

Learning effects generally occur because of learning in the literal sense a n d /o r innovation. For th is reason, learning effects are more extreme in operations m ade up of more worker assem bly th a n m achine

3The learning index is the ratio of the logarithm of the learning rate to the logarithm of 2 and is, therefore, always less th a n or equal to 0.

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time and in swiftly growing or improving industries.

The docking facility u n d e r evaluation here leans toward assem bly v ersu s m achine time b u t is n ot subject to frequent innovation. The potential for the learning effect is present b u t n ot guaranteed.

In the succeeding chapters of th is paper we will deal w ith the M /G /1 and the M /M /s queues, both with potential for learning effects, w ith the goal of developing a cost minimizing design.

C hapter 2 will derive the m ean num ber of custom ers in the queue (the previously m entioned L) for both the queues. A proof of single server optim ality in th e M /M /s queue will also be presented an d th u s simplify the optim ization of th a t queue. C hapter 3 will describe the geometric program m ing m ethod used in the actual optimization.

C hapter 4 will ru n through sam ple problem s and offer verification of the optimization results. C hapter 5 will conclude the paper with some discussion and recom m endations for further research.

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Chapter 2

DERIVATIONS FROM QUEUING THEORY

This chapter will specialize the Hillier cost model for more particular queue system s beginning with the single server queue.

Recalling th a t the model is of the form

TEC = s C J { \i) + CWL,

the only variable dependent upon th e queue type is L, the m ean num ber of custom ers in the system . Since the goal is to minimize w ith respect to p, the m ean service rate, L m u st be expressed in term s of p. X, the m ean arrival rate, and th e variance of the service time distribution, b(t).

Single Server Q ueues

For the single sever queue with Poisson in p u t (i.e. exponentially distributed interarrival times), it is possible to derive an expression for L in term s of only these basic param eters. Let p represent the ratio of X to p which is referred to as the traffic intensity or utilization factor for a single server queue.

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upon the num ber of custom ers waiting a t the preceding time n.

Consider the queue system ju s t following a custom er departure. Here Xn indicates the num ber of custom ers waiting a t time n and A represents the n u m b er of arrivals during the p a s t service period of length t.

f X -1+A X > 0 X a {

' A Xn=0 ,

The expression is easier to deal with algebraically w hen rew ritten as:

In the steady state situation, th e system state, a t time n, is

essentially independent of the initial state of the system (time zero). The num b er of custom ers a t any time n, is a random variable with a single m ean and variance regardless of the value of n. T hat is,

where the superscript on L indicates the evaluation takes place a t the m om ent directly after a departure.

V d - x » - u ( x » > + A ,

1

x

>0 where U(X„) = { „ X" =Q . n (2.1) £(X„.1)=£(X„)=L«> (2.2)

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Taking the expected value of eq 2.1 and using eq. 2.2:

£[U(Xn)]=£G4).

The expected value of A can be calculated utilizing a Riemann-Stieltjes integral:

E{A) = (E[A\T=t]3B(i) = f \ t 3 B ( t ) = A.£[f] = A./M- = P

Squaring eq. 2.1:

= K + [U (X „)]2 + A 2 + 2AX„

- 2A[U(X„)] - 2 [U (X n)]X„. Again, invoking the equilibrium conditions.

O v - O Y

A»l *11 + 1

+ = E ^ + E ^ f then

E(X-h) = £(X 2„). Also, it is intuitively obvious th a t

[U (X „)]2 = U (X „),

a n d

(2.3)

X .[U (X .)] = X ..

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the nu m b er of arrivals is independent of the num ber of custom ers in the system (i.e. A is independent of X J, resu lts in the following:

2 L(D) - 2L (D)p = E{A2) + p - 2 p 2,

£,C°) = P ~ 2 p 2 + E(A2) (2.4)

2(1 - p)

Of course,

E(A2) = var(A) + E(A)2 = var(A) + p2. ®.5)

But, recalling th a t t is the length of the preceding service period, th e variance of A can be expressed as follows:

w (i4 ) = E[var{A | r = f ) ] + var[E(A \ T=t)]. (Parzen, p.55) Where, E(A\T~t) = £ a e - u ( X t Y = = Uy o i ( a - l ) l and £ [ y l ( i 4 - l ) | r = J ] = J ' o o! ,-kts 1 *\a-2 = at)< " = A2f2. 2 (0-2)! Then,

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var ( A\ T=t ) = E [ A ( A - 1) | T=t] + E ( A \ T = t ) - [E(A \ T - t ) ] 2

= X2t2 + Xt - X2t2 = Xt. So,

var{A) = E{Xt ) + var(Xt ) = 12£(f) + l 2v cr(0 = p + X2var(t). ® .6) The variance of t is, in th is case, the variance of the service times; th a t is, var(t) = var(b(t)).

Finally, com bining eq.s 2.4, 2.5, and 2.6, we arrive a t the Pollaczek-Khinchine formula:

£(°) = p - 2 p 2 + p + p2 + X2var(t) 2(1 - p)

= | p2 +A,2vgr(r) 2(1 - p)

It rem ains now to prove th a t the m ean n u m b er of custom ers in the queue the m om ent after a served custom er departs is equivalent to the m ean num b er of custom ers in the queue a t any given m oment.

It is tru e for any Poisson process, th a t is, any queue with in p u t corresponding to a Poisson distribution, th a t th e m ean num ber of custom ers imm ediately prior to a custom er arrival is equivalent to the m ean nu m b er of custom ers a t an y given m om ent (see Cooper p .57 for a concise proof). The hypothesis th a t th e m ean ju s t after a departure is the sam e as the m ean ju s t prior to an arrival is logically equivalent to

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the hypothesis th a t dn, th e steady sta te probability of n custom ers in the system ju s t after a departure, is equal to a t h e steady state probability of n custom ers in th e system ju s t prior to an arrival.

For proof of this second hypothesis let AJt) denote the nu m b er of arrivals, during the time interval (0 ,t), th a t increase the n u m b er of custom ers in the system from n to n + 1 , and Dn(t) denote the nu m b er of d ep artu res on the sam e interval th a t decrease the num ber of custom ers in the system from n+1 to n. Then, considering a long interval of length

T’ \An(T )-D n(T)\ s 1. (2.7)

Since a steady state solution is assum ed to exist (i.e. X/\x < 1), the limit of the ratio of the total num b er of departures in time T, D(T), to the total num ber of arrivals in tim e T, A(T), m u st approach 1 as T goes to infinity:

UmDOO _ j (2 .8)

t^ A ( T )

Dividing eq. 2.7 by A(T) and taking the limit as T approaches infinity:

lim _ D nW | ^ lim 1 _ ~

r-ool A ( r ) I r-oo^CD = Then, using eq 2.8,

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lim An(T) _ lim T - ~ M T ) = t ^°°A(T)

lim Dn(T) lim A(T) " t ^°°A(T) t -*°°D(T) lim ^ n(^ ) £)(7) • But, lim _ lim _ . T~**° A(T) ~ a" T" co D(T) ~

(See Gross and Harris, p 235) Hence, th e probability of n custom ers in the system prior to an arrival is equal to th e probability of n custom ers ju s t after a departure. From th is it follows th a t th e expected n u m b er of custom ers, ju s t prior to a n arrival and ju s t after a departure are equal. Finally, since the M /G /1 queue arrival tim es are Poisson distributed, L(D), the expected num ber of custom ers in the system after a departure, is equivalent to L^, the

expected num ber of custom ers in the system prior to an arrival, which is equivalent to L, the expected num b er of custom ers a t any time.

There are, of course, m any types of single server queues with Poisson input.

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l - p ii-A.

For an M /D /1 queue: var(s)=0, and

L = p ♦ — B i - =

2(1 - p ) 2 ( 1 — p )

The M /G /1 resu lts are, however, more convenient for u se in general evaluations and calculations as will be performed in ch ap ter 3.

A M ulti-server Queue

Deriving expressions for m ulti-server queues is m uch more

difficult, and, for th a t reason, only the M /M /s queue (i.e. the queue with exponential interarrival tim es, exponential service tim es, and s parallel servers with a single queue) will be attacked here. To begin, the

probability th a t there will be n custom ers waiting a t time t + 8t is the sum of five other probabilities:

(1) th e probability of n custom ers a t time t with no custom ers arriving or departing during tim e 8t; (2) the probability of n custom ers a t time t with both an

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arrival and a departure during time 8t;

(3) the probability of n+1 custom ers a t time t with one ~~ departure and no arrivals during 8t;

(4) the probability of n - 1 custom ers a t time t with one arrival and no d ep artu res during 8t;

(5) the probability of any other initial num b er of custom ers a t time t requiring more th a n one

departure or arrival during 8t in order to resu lt in n custom ers a t tim e t+8t.

Since the inter-arrival tim es and service tim es of th is type of queue are given as exponentially distributed random variables with param eters X and p, respectively (where X still denotes the m ean arrival rate an d p the m ean service rate), the num ber of custom ers arriving or being served during any time interval of length t are random variables following

Poisson distributions with param eters Xt and pt. (for proof see F reund and Walpole p. 211). T hat is,

( X b t) n e~xdt

Prob(n arrivals on interval bt) - --- . n\

Similarly,

ARTHUH LAKE I LIBRARY COLOMBO SC3SOOL o& MINES

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e~*6t Prob(m departures on interval 6 0 = — — --- .

m ! Using the M acLaurin’s series expansion of e'x5t:

+ - a * # + ... + t » nW ' +....

2 6 n 1

th e probability of any n u m b er of arrivals or d epartures during the time interval St can be derived. For notational consistency let n denote the n u m b er of arrivals an d m the n u m b er of departures.

Prob(n-Q) - 1-A.Sf + o (6 0 , P ro b (n -l) = A.61 + o(&t), Prob{n>\) = o(S0, Prob(m=Q) = l - | i 6 f + o(60» P rob(m -1) = |i6r + o (6 0 , Prob(m> 1) =

o(60-o(8t) represents a su m of term s involving greater th a n first power term s of St. More precisely, o(8t) signifies any term or sum of term s f(St) su c h th a t

Km / ( S O _ n 6M>"~sT "

Note th a t as St becom es sm all the probability of m ultiple arrivals (n > 1) or d epartures (m > 1) on a single interval becom es negligible.

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Now letting Pn(t) denote the probability of n custom ers a t tim e t and using the probabilities above the following recursive expression can be written:

+J\_i(f)[(*ar+O(ao)(l - ii8r+o(8f))]

^ . ♦ 1( 0 [ ( i * a / + o ( d O ) ( i - A ^ + o ( 5 0 ) ] +E P,.X f)[o(8r )]+£ j>ll.X0[o(ftf].

- « 2

This equation yields different resu lts w hen n = 0, n > s, and n < s. Taking the limit as 8t - » 0 (at th is point all o(8t) term s will go to 0)

generates the following three equations:

dPJt) - g - i = -AP0(t) + n P ^ t ) , dP (t) = -(A. + « | i) P ll( 0 + A P , ,. ^ ) + ( n + l ) | i P atl( 0 fo r 1 u s s - 1 , at dp m - " — = - ( A + s n ) P „ ( 0 + AP ,.,(() + 5(iP„^(0 fo r s a n . at

As in the Single server case, a steady state solution for L is w hat is desired. The steady state requires th a t the probability of any nu m b er of custom ers being in the queue be co n stan t over time. T hat is,

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M . , . dt Thus, a t equilibrium, p, = A p 0 = pPo, kPn_i = (k+n\i)PH - (n + 1) f or 1 u u - 1 , x p n-1 ■ - (5)H^„+i for s * n.

In these equations, however, p can no longer be considered the traffic intensity or utilization. T hat label is reserved for the value p = ^/(p*s). Here p is a direct su b stitu tio n for the value X /\if and h a s no other real significance.

From here it can be proven by induction th a t

Pn = - L p ’P0 f o r m s , s*-"

P , = — p nP0 for s < n . SI

Using the fact th a t the sum of the probabilities m u st be 1, P0 can be derived,

*0 - o * * ? ( — )]'1

11 s\ s - p

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V'*1

o ( s - O K s - p )2

This equation, as well as those derived for the single server queues can now be inserted into the original cost model for any chosen value of s, u n d er any learning effect condition, and subsequently, the queue can be optimized. However, during the coding and testing of the algorithm described in chapter 3, it w as realized th a t for the M /M /s queue to be operating optimally, s m u st equal one. The proof of this lies in

com paring the cost of the M /M /s queue with m ean service rate of \i and the cost of the M /M /1 queue with m ean service rate sp. The two

system s have equivalent servicing capability b u t very different costs.

Where s is an integer greater th a n zero, p is the ratio of X to p and is bounded between zero and s by the equilibrium condition, and m is a

Given:

C n 2 Cost (M/M/1,s\i) = Cs(\is)m + Cj> + — and

Cost (M/M/s,\i) = Css\Lm + C^p +

( s - i ) i ( 5 - p ) 2r £ - * p ’{s-p) 0 ll

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real num ber less th a n or equal to 1. Then:

Cost (M/M/1,s\i) ^ Cost (M/M/s, \i) implies

0 k Cost (M/M/1,s\i) - Cost (M/M/s, ji) Which, in tu rn , implies

0 k CsKmp " " (s m- s ) + C X ^ - l ) + S ^ . \ p --- E l— --- ]

* s - p L S *_1 71i 1

( s - l ) \ ( s - p ) 2 'L ^ - + p 5 o t!

Which, again, implies

(s -p )C sk m( s - s m) p*

--- * 1 - s + p - r

C D m + 1 * ~1n i

^ + p*

o i!

W hen s is one, the two queues are the sam e, so the two costs are obviously equivalent. W hen s > 2, p < 1 implies s > 1 + p. Knowing this, the left-hand side m u st always be positive, the right h an d side always negative an d th e single server cost is always less th a n th a t of the m ulti­ server queue.

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th a t it m u st be strictly greater th a n X. In th e shipping problem studied here, p is adjusted by increasing manpower, upgrading and improving m achinery, increasing th e num b er of accesses (ramps, cranes, etc.) to the ship being serviced, etc. In the case of a high ratio of waiting costs to service costs, the optimal m ean service rate m ight climb so high as to be impossible with a single server in a real situation (i.e. a situation w here p is bounded).

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Chapter 3

GEOMETRIC PROGRAMMING AND THE SINGLE SERVER QUEUE

Now there are complete cost models in term s of the co n stan ts— service cost per custom er per server, C8, waiting cost per custom er per u n it time, Cw, m ean arrival rate, X, num ber of servers, an d variance of service tim es, var(t)—and dependant upon a single variable, p, the ratio of the m ean arrival rate to the m ean service rate.

For the M /G /1 queue:

and for the M /M /s queue, which is optimal w hen s = 1 (by the preceding proof, pp. 27 and 28):

The ta sk a t h an d is to minimize these cost equations with respect to the ratio of m ean arrival rate to m ean service rate, p.

CwX2var(t)

+ 2(1 - p)

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Geom etric Programming Overview

Geometric program m ing (GP) is a still evolving technique for

m inim ization of linear and nonlinear system s. The technique, which w as first studied by Duffin, Peterson, and Zener in the 1960’s, is derived from C auchy’s arithm etic-geom etric inequality and the fact th at, a t optimality, the inequality becom es a n equality ( a more complete history of GP is contained in Thome (1988)).

GP revolves ab o u t four basic rules, which are described as follows by Woolsey and Sw anson (1969):

Rule 1: The form of th e optim al solution of any posynomial GP problem is:

TEC* = II (Coefficient of term if b t)

objjimcL

E x II [ II (Coefficient of term j/b j) x ( E b j f " ^ ] ,

aUeontL aeonsf. aconst.

where the asterisk indicates a value a t optimality, i refers to a term in the objective function, and j refers to a term in a constraint. Rule 2: The exponent m atrix is constructed as follows:

Rule 2A: The sum of the contributions to cost in the objective function is one.

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E b t = 1 .

oh], fund.

Rule 2B: For each prim al variable the equations in the exponent m atrix are:

E (5, exponent o f variable at termi)

ohjjunct.

+ E (b j exponent o f variable at termj) = 0 .

allconst.

Rule 3: At optimality, for each term in the objective function

TEC* tertni

~ & r'

Rule 4: At optimality, for every term , in each constraint

b k - termk ( E bp .

const.

These four rules (the first two of which con stru ct the dual problem while the la st two dem onstrate relationships a t optim ality th a t facilitate easier solution) allow direct solution of simple problems.

Degree of difficulty (hereafter, DD) is defined as the num ber of term s in both the objective function and the constraints, m inus the n um ber of term s in th e problem, m inus one. Zero DD problem s are

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easily solved using th e four basic rules. Higher DD problems, however, require modification before they can be optimized.

Ratliff (1986) developed a m ethod to optimize nonlinear, multi- variable, unconstrained posynomials. By holding all variables b u t one constant, condensing all like term s, and iterating until the variable settled on a value, th e n moving to another variable and cycling through all the variables u ntil they all settled, Ratliff arrived a t the optim um of high DD problems.

Later, Thome (1988) used the Greening technique to optimize a class of nonlinear, single variable, unconstrained signomials. This class of problem s w as restricted to term s with positive coefficients, or negative coefficients and positive exponents. By putting his problem s into a form su ch th a t the objective function becam e a constraint on an additional variable and m anipulating th is constraint until the negative coefficient term s could be condensed with th is new variable, he w as able to condense the entire problem into a zero DD problem and optimize.

A third option for using GP on high DD problem s is to select from the original problem some term s with which to build a lower DD,

bounded sub-problem an d solve th a t sub-problem to obtain bounds on the actual objective function and the variables.

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If the M /G /1 cost model is reform ulated to the stan d ard GP form (where the objective function consists of term s with variables, exponents, and coefficients, versus term s w ith polynomial expressions, exponents, and coefficients), a su b stitu tio n for the awkward denom inator in the waiting cost m u st be made, creating a constraint.

TEC = CsXm p - + Cw p + CwpaY-1 + Cwy~l \ 2var(t)

s.t. p + y s 1 .

The problem now h a s six term s, two variables, and three degrees of difficulty. A sim ilar reform ulation can be found for the M /M /1 queue, with four term s, two variables, and one DD; The constraint elim inates the possibility of using either Thome’s or Ratliffs method, so, perhaps bounding is the only option left (using GP) to evaluate th is problem.

D istorting GP to Solve th e Single Server Queue

Up until now, condensation of term s h as been performed only on like term s (i.e. on term s w ith the sam e sign exponent and the sam e sign coefficient). This is because condensation of like term s yields a like term , while condensation of dissim ilar term s yields a term which is not wholly predictable. Single server cost models, however, can be

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reform ulated so th a t condensation of the objective function yields a single term , for which the signs of th e exponent and coefficient are predictable.

To arrive a t a zero DD, single server cost problem, all the term s of the objective function m u st be condensed (since there are two variables an d two term s in th e constraint). T hat is, the problem m u st appear as follows:

TEC =

*pwV 2

s.t. p + y *1 .

For the above problem to balance and be bounded, both ©1 an d ©2 m u st be less th a n zero. So, a form of the objective function m u st be found which condenses to the form above su ch th a t ©1 and ©2 are always negative in the solution space 0 < p, 0 < y.

Through necessity, ingenuity, trial and error, etc., a form of the M /G /1 cost model fulfilling those requirem ents w as found. Since

I - P2 + + var{t)X2 _ _P + 1 1 + var(t)X2

= 2 ( 1 — p ) + P + 2 ( l - p ) = 2 + 2 p ( l - p ) ' 2 p ' 2 + 2( l - p ) = ,5p + .5p'*Y_1 - .Sp'1 - .5 + var(t)^

2(1 — p ) the objective function can be reform ulated as

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TEC = Cgk mp-m + J5Cwp-ly-1 ~ -5Cwp '' + .5Cwp + .5CwY_1v ar(0^2 - *5CW Ignoring the constant, moving the signomial term to the left h an d side, and condensing yields:

fTEGy* 0 / yft8 / ^wP v>4 .6, ' V 2 p 5 j p " « 2 2 p Y63 2 Y85 72TC .5C p _1 where 50 = --- — --- = 7EC + .5Cwp 1 7EC + .5Cwp * = C,X” p ” = SC w Y 'V 1 -l 2 7EC + .5Cwp 1 3 TEC+ ,5Cwp~1 3 - -5CwP = -5Cwvar(t)X2y~1 4 ’ TEC + .5Cwp ' 1 5 ’ TEC+.5Cwp"1 Then, TEC = *pwlYw2 s.f. p + y s 1 , (3.1) * * * * + .5C ( p ! ) ]

where

o )i. -»»» ^ 8< +

\ : . h

. — --- --- !—

lz £ .

...

60 TEC- ( « , + 8,) -(•5cwy '1p‘I + J C . m f O X 2^ ) co 2 = --- = --- — ---50 TEC ». 2 6 , j- C l * t C„ £ C 'j1 C var(,t)X2 £ * = 80 (-pr-) ( - 7— ) ( r f > ( r f - ) 0 ( * ) ° • Cw o2 2o3 2o4 2o5

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and y are positive (recalling th a t if y is positive th e n p is < 1), so the condensation was successful.

Now the four GP rules can be used to solve the problem directly. By rule 2: dx = i c + d3 = 0 0)2^ + ^ = 0 . S o, = 1 , 5 2 = - 0) 1 , d3 = -co 2 . Then by rule 1: TEC' = = ifc(-u2)"2(-G>l)“1(-w 2 - u i r <'’2-"1. <3 -2> B ut by rule 4:

= -o)2 = 'i{d 2 + 4?) 55 “Y(o>1 + o>2)

= -o)l = p(dj + ^3) = —p ( to 1 + 0)2) . So,

- 0)2

Y = r P -- 0)1

S ubstituting th is for y in the original equation [3.1]:

TEC* = k p wl + w2(-0)2)<,>2(-0)1)“<,)2 . (3 *3) Solving eq.s (3.2) and (3.3) together:

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i k C - w a ^ t - o i r ^ ^ - w l ) - * 2-"1 = * p wl + • 2( - « 2 ) * 2( - » l ) " * 2 col

p = --- • col + co2

Iteration through these equations, from any initial p, appears to converge quadratlcally to the optim um ju s t as with Thome’s and R atliffs m ethods. Once p* h a s been found, p* = X/p* follows directly and the optimal m ean service rate is determ ined.

The flow ch a rt for th is algorithm (appendix A) attem pts to control round-off error by avoiding division. T hat is, since the value of p derived finally from the above equations can be reduced to a ratio of term s from the objective function, the program simply defines five variables to be the five term s of the objective function, two variables to be the nu m erato rs of the co’s, and the new value of p to be the ratio of the first num erato r to the sum of the num erators. In th is way, propagation of round-off error is avoided, while the final resu lt for each iteration rem ains the sam e.

A sim ilar derivation h a s been performed on the M / M / 1 queue with the following results:

L = - 2 - = --- --- - 1 1 - p p ( l - p ) p

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C C TEC = Cg\ mp~m + - JL - -J* - Cw s.t. p + y * 1 PY P TEC = * p wV 2 - * C , A - p - - Cw( - i - > _c , , w te r e o> 1 = ---— o)2 = — ---TEC TEC 0)1 P = —* 0)1 + 0)2T •

Again, col and co2 are negative wherever y an d p are positive so the condensation is successful an d will converge on th e m inim um . The flow ch art for th is algorithm (appendix A) also u ses th e m ethod discussed above to avoid round-off error.

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Chapter 4

SAMPLE RUNS OF THE ALGORITHM AND EXPLORATION OF THE

QUEUE MODEL

It is not the in ten t of th is paper to present more th a n empirical proofs of either the geometric program m ing algorithm s’ optim ality or their rate of convergence. To dem onstrate both the success of the optim ization m ethod an d interesting aspects of the two models, five example problem s for each of the two queue types, M / G / l an d M /M /1, (see Table 4.1) have been chosen.

E ach of these models w as ru n with nine different startin g points on the open interval (0 , 1) (p is a probability, and as su ch is restricted to the closed interval [0 , 1]; the endpoints, zero and one, lead to division by zero error, so th e open interval w as used).

The iteration criterion w as the absolute value of the difference of the two m ost recent p’s calculated. If this absolute value exceeded 10'5, the iteration would continue.

Verification of the algorithm s’ results, in the form of spreadsheets of total expected cost on the interval 0 < p < 1, is found in appendix B.

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Table 4.1: Example Problems Used for Analysis Problem Mean Arrival Rate X Cost of Waiting

cw

Cost of Service

c.

Learning Rate m Variance var(t) M /M /1 A 2 5 10 1 B 2 5 10 .87 C 1 2 0 .5 .81 D 1 .5 20 .81 E 2 1 3 .93 M /G /l A 1 4 4 .95 .7 B 4 1 20 .97 .7 C 4 2 0 1 .9 7 .7 D 2 1 3 .93 0 E 2 1 3 .93 .1

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The M /M /1 Examples

Since the M /M /1 queue is actually a special case of the M / G / l , these problem s will be discussed first. The five M / M / 1 problem s were chosen to illustrate the effects of learning on the total cost and the responses (in cost) to extreme differences between service cost per custom er and waiting cost per custom er.

The first problem (M /M /1A in table 4.1) is one with no learning effect and is modelled by the following TEC equation:

This w as verified by the algorithm (or vice versa), as seen in table

+ L—

P 1-P

This problem Is simple enough to solve w ith calculus:

§ T U T -OC\ 5

implies

1.5p2 - 4 p + 2 = 0 .

Then, by the quadratic formula:

2 ******* 1 ZilNEs >0 80401 ^OLOfy|,Do gOLDBfif' q M T H O M l a e s s

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4.2. Convergence (to five decimals in p and eight decimals in the total expected cost), for all nine starting values, w as achieved in ten to twelve iterations.

Table 4.2: R esults o f Problem M /M /1A

Initial p Pinal p P Total Exp. Cost Iterations .00001 .6 6 6 6 6 9 2 .9 9 9 9 9 4 0 .0 0 0 0 0 0 0 0 1 3 12 .001 .6 6 6 6 6 9 2 .9 9 9 9 9 4 0 .0 0 0 0 0 0 0 0 1 3 12 1 .6 6 6 6 6 0 3 .0 0 0 0 3 4 0 .0 0 0 0 0 0 0 0 9 2 11 10<N.6 6 6 6 6 6 3 .0 0 0 0 2 4 0 .0 0 0 0 0 0 0 0 5 9 11 .5 .6 6 6 6 6 4 3 .0 0 0 0 1 4 0 .0 0 0 0 0 0 0 0 1 3 11 .75 .6 6 6 6 6 2 3 .0 0 0 0 2 4 0 .0 0 0 0 0 0 0 0 4 1 10 .9 .6 6 6 6 7 1 2 .9 9 9 9 8 4 0 .0 0 0 0 0 0 0 0 4 6 11 .999 .6 6 6 6 7 4 2 .9 9 9 9 7 4 0 .0 0 0 0 0 0 0 1 1 4 11 .9 9 9 9 9 .6 6 6 6 6 4 3 .0 0 0 0 1 4 0 .0 0 0 0 0 0 0 0 1 3 12

Problem M / M / IB adds an 87% learning rate to the first problem resulting in the following TEC equation:

TEC = 34,82 + .

p ‘J ! - P

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explained in chapter 1. The GP resu lts reflect this (see Table 4.3),

converging, in seven or eight iterations from all startin g points, on a TEC nearly eighteen percent less th a n the optimal total expected cost w ithout learning.

Table 4.3: R esults of Problem M /M / IB

Initial p Final p P Total Exp. Cost Iterations .0 0 0 0 1 .6 1 3 8 2 8 3 .2 5 8 2 4 3 3 .6 7 4 4 5 0 0 2 6 6 8 .001 .6 1 3 8 2 9 3 .2 5 8 2 4 3 3 .6 7 4 4 5 0 0 2 5 0 8 .1 .6 1 3 8 3 2 3 .2 5 8 2 2 3 3 .6 7 4 4 5 0 0 2 2 1 7 .25 .6 1 3 8 2 8 3 .2 5 8 2 4 3 3 .6 7 4 4 5 0 0 2 7 1 7 .5 .6 1 3 8 3 1 3 .2 5 8 2 2 3 3 .6 7 4 4 5 0 0 2 2 8 7 .75 .6 1 3 8 3 9 3 .2 5 8 1 8 3 3 .6 7 4 4 5 0 0 2 5 1 7 .9 .6 1 3 8 3 2 3 .2 5 8 2 2 3 3 .6 7 4 4 5 0 0 2 1 9 8 .9 9 9 .6 1 3 8 3 2 3 .2 5 8 2 3 3 3 .6 7 4 4 5 0 0 2 2 8 8 .9 9 9 9 9 .6 1 3 8 3 1 3 .2 5 8 2 3 3 3 .6 7 4 4 5 0 0 2 2 8 8

Problems M / M / 1C and M / M / I D were chosen to dem onstrate the stronger effect on total cost of service cost per custom er th a n waiting cost per custom er.

M /M /1C h a s an extremely high waiting cost (relative to service cost) of 20 m onetary u n its per custom er per u n it time, resulting in the

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following cost equation:

and yet the optimal total cost is only 4.67 (see Table 4.4). Table 4.4: R esults o f Problem M /M / 1C

in itial p Pinal p P Total Exp. Cost Iterations .00001 .0 8 3 7 6 0 1 1 .9 7 9 4 9 4 .6 6 5 2 3 2 7 7 1 7 56 .001 .0 8 3 4 7 9 1 1 .9 7 9 0 3 4 .6 6 5 2 3 2 6 5 7 0 55 .1 .0 8 3 6 0 8 1 1.96061 4 .6 6 5 2 3 2 5 8 2 5 3 6 .25 .0 8 3 6 1 3 1 1 .9 5 9 8 2 4 .6 6 5 2 3 2 7 7 7 9 4 6 .5 .0 8 3 6 0 4 1 1 .9 6 1 0 9 4 .6 6 5 2 3 2 4 7 1 7 4 9 .75 .0 8 3 6 1 1 1 1 .9 6 0 1 5 4 .6 6 5 2 3 2 6 9 4 0 4 9 .9 .0 8 3 6 1 3 1 1 .9 5 9 8 2 4 .6 6 5 2 3 2 7 7 8 4 4 9 .9 9 9 .0 8 3 6 0 4 1 1 .9 6 1 0 9 4 .6 6 5 2 3 2 4 7 1 4 50 .9 9 9 9 9 .0 8 3 6 0 4 1 1 .9 6 1 0 9 4 .6 6 5 2 3 2 4 7 1 6 50

M / M / I D h a s a service cost of 20 m onetary u n its per custom er and total cost equation as below:

TEC = — +

p - 7 2 ( 1 - p )

Which leads to a optim al total cost of approximately 25.22 (see Table 4.5). This indicates th a t the optim al TEC is m uch more sensitive to the

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Table 4.5: R esults o f Problem M /M / ID

Initial p Pinal p P Total Exp. Cost Iterations .00001 .8 3 7 4 7 1 1 .1 9 4 0 7 2 5 .2 2 0 2 7 0 0 6 6 2 25 .001 .8 3 7 3 6 1 1 .1 9 4 0 9 2 5 .2 2 0 2 7 0 0 6 7 0 2 2 .1 .8 3 7 4 7 2 1 .1 9 4 0 7 2 5 .2 2 0 2 7 0 0 6 6 4 2 4 .25 .8 3 7 4 7 2 1 .1 9 4 0 7 2 5 .2 2 0 2 7 0 0 6 7 2 2 4 .5 .8 3 7 4 7 1 1 .1 9 4 0 7 2 5 .2 2 0 2 7 0 0 6 5 9 2 4 10.8 3 7 4 7 1 1 .1 9 4 0 7 2 5 .2 2 0 2 7 0 0 6 6 1 22 .9 .8 3 7 4 6 1 1 .1 9 4 0 9 2 5 .2 2 0 2 7 0 0 6 6 5 22 .9 9 9 .8 3 7 4 7 1 1 .1 9 4 0 7 2 5 .2 2 0 2 7 0 0 6 5 9 25 .9 9 9 9 9 .8 3 7 4 7 1 1 .1 9 4 0 7 2 5 .2 2 0 2 7 0 0 6 5 9 2 5

service cost per custom er th a n to waiting cost per custom er.

The final example of th is type problem, M / M / I E , m ay be viewed as a typical problem reflecting the original shipping model discussed. In th is example, the waiting cost per custom er is one third the service cost per custom er. This reflects the n atu re of the shipping dock since the ships (customers) are owned by the dock operator, implying the waiting cost is due only to the ships’ lost h o u rs of productivity. The ninety percent learning rate is reasonable in su ch a labor intensive operation, and, if the u n it of time is considered a day or work week, two arrivals during th is time is appropriate (per week if the ships are large, per day if

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small).

Given the param eters as in table 4.1, th e re su lta n t model is of the form:

As illustrated in table 4.6, the optimal ratio of m ean arrival rate to m ean service rate w as found to be approxim ately .68 , with a m inim um total expected cost of operation of approxim ately 10.04 m onetary u n its per time unit.

Table 4.6: R esu lts o f Problem M /M / IE

In itial p Pinal p P T otal Exp. Cost Iteration s .0 0 0 0 2 .6 8 7 8 0 1 2 .9 0 7 8 2 1 0 .0 4 3 3 6 8 0 1 4 5 11 .001 .6 8 7 8 0 2 2 .9 0 7 8 1 1 0 .0 4 3 3 6 8 0 1 5 3 10 .1 .6 8 7 7 9 8 2 .9 0 7 8 3 1 0 .0 4 3 3 6 8 0 1 3 4 12 .25 .6 8 7 7 9 8 2 .9 0 7 8 3 1 0 .0 4 3 3 6 8 0 1 3 3 12 .5 .6 8 7 7 9 0 2 .9 0 7 8 7 1 0 .0 4 3 3 6 8 0 1 4 0 11 .75 .6 8 7 7 8 9 2 .9 0 7 8 7 1 0 .0 4 3 3 6 8 0 1 4 4 10 .9 .6 8 7 7 9 2 2 .9 0 7 8 6 1 0 .0 4 3 3 6 8 0 1 3 3 12 .9 9 9 .6 8 7 7 9 0 2 .9 0 7 8 7 1 0 .0 4 3 3 6 8 0 1 4 0 12 .9 9 9 9 9 .6 8 7 7 9 0 2 .9 0 7 8 7 1 0 .0 4 3 3 6 8 0 1 4 0 12

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The M /G /l Exam ples

The M /G /1 problem s were chosen to dem onstrate th e effects of interarrival time variance (previously referred to as var(t)) on the optimal total expected cost, and again, the strong dependence of the optimal total cost on the service cost per custom er.

M /G / 1A is of th e form:

TEC = —— + *

'11

.

p - * 2 ( 1 —p )

This model was chosen because of the equality of the service and waiting costs per custom er. The optimal ratio of m ean arrival rate to m ean service rate is approxim ately .49 (see table 4.7). This resu lt is interesting to com pare to the resu lts of problems M /G /IB and M /G / 1C (see tables 4.8 and 4.9, respectively).

Example M /G / IB is sim ilar to M /M / ID in th a t the service cost per custom er is considerably greater th a n the waiting cost per custom er. The cost equation is of the form:

TEC = + p + *7

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Table 4.7: Results of Problem M /G /1A

Initial p Final p P Total Exp. Cost Iterations

.00001 .486953 2.05359 13.4118065116 10 .001 .486952 2.05356 13.4118065120 10 .1 .486953 2.05358 13.4118065116 11 .25 .486961 2.05355 13.4118065129 10 .5 .486952 2.05359 13.4118065119 8 .75 .486950 2.05360 13.4118065132 10 .9 .486958 2.05356 13.4118065116 11 .999 .486959 2.05356 13.4118065117 11 .99999 .486959 2.05356 13.4118065117 11

Table 4.8: R esults of Problem M /G /IB

Initial p Final p P Total Exp. Cost Iterations

.00001 .771701 5.18330 122.0847497261 192 .001 .771698 5.18337 122.0847497247 193 .1 .771708 5.18331 122.0847497246 192 .25 .771698 5.18337 122.0847497258 189 .5 .771708 5.18330 122.0847497254 182 .75 .771708 5.18330 122.0847497254 144 .9 .771708 5.18331 122.0847497253 179 .999 .771708 5.18331 122.0847497250 193 .99999 .771708 5.18331 122.0847497252 193

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Table 4.9: Results of Problem M /G /1C

Initial p Final p P Total Exp. Cost Iterations

.00001 .13603 29.40557 157.4004681453 11 .001 .13602 29.40815 157.4004681084 10 .1 .13603 29.40634 157.4004680913 11 .25 .13602 29.40648 157.4004680855 11 .5 .13603 29.40577 157.4004681280 11 .75 .13603 29.40557 157.4004681452 11 .9 .13063 29.40564 157.4004681389 11 .999 .13603 29.40575 157.4004681295 11 .99999 .13603 29.40575 157.4004681294 11

Models M /G /1C and M /M /1C are sim ilar in their high waiting costs per custom er relative to service costs per custom er. Model M /G /1C h a s a total expected cost equation of:

TEC = + *>(P + .7) p-*7 2(1 - p)

The resu lts of these two models illustrate the sam e effects seen in th e M /M /1 examples: w hen the service cost per custom er is high relative to the waiting cost per custom er, the optimal ratio of m ean arrival rate to m ean service rate is high (In the case of M /G /B , p * .77 and for

M /M /ID , p « .84); w hen the service cost per custom er is equal, or nearly equal, to the waiting cost per custom er, the optim al ratio will tend to one

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half (as in M /G / 1A); and w hen the service cost is relatively small, the optim al ratio will be sm all (For M /G /1C, p * .14 and for M /M /1C, p * .08).

Problems M /G /ID and M /G /I E are two other versions of the typical shipping example introduced as M /M /IE . M /G /ID is a determ inistic model of the form:

__ _ 5.60 p2

TEC - ——- +

p - 9 2(1 - p)

M /G /I E is the probabilistic version of the problem, assum ing a variance of . 1, so th a t the model becomes:

T E C = 5 * 0 + p2+.l p-» 2 ( 1 - p )

As m ight be expected, the optimal total expected cost of the

determ inistic model (see Table 4.10) is less th a n th a t of the probabilistic model (see Table 4.11). The exponential service model of th is problem (M /M /IE) is even slightly more expensive because the variance of

exponential service is the inverse of the square of the m ean; in th is case slightly less th a n . 12.

Beyond the interesting aspects of each of these ten particular models, it is im portant to note the convergence of the algorithm s. Not

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Table 4.10: Results of Problem M /G /ID

Initial p Final p P Total Exp. Cost Iterations

.00001 .752333 2.65840 9.1273515668 13 .001 .752335 2.65839 9.1273515661 17 .1 .752333 2.65840 9.1273515671 17 .25 .752333 2.65840 9.1273515669 17 .5 .752335 2.65839 9.1273515662 17 .75 .752344 2.65836 9.1273515668 10 .9 .752342 2.65837 9.1273515662 17 .999 .752335 2.65839 9.1273515662 18 .99999 .752335 2.65839 9.1273515662 18

Table 4.11: R esults of Problem M /G / IE

Initial p Final p P Total Exp. Cost Iterations

.00001 .719739 2.77879 9.8840006268 18 .001 .719738 2.77878 9.8840006267 20 .1 .719740 2.77882 9.8840006264 21 .25 .719728 2.77878 9.8840006273 20 .5 .719728 2.77883 9.8840006275 19 .75 .719729 2.77883 9.8840006275 16 .9 .719738 2.77879 9.8840006275 20 .999 .719738 2.77879 9.8840006267 21 .9999 .719738 2.77879 9.8840006267 21

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only was the optim al cost calculated to extreme precision but, this resu lt w as achieved, with only two exceptions, in less th a n 25 iterations.

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Chapter 5

CONCLUDING STATEMENTS AND RECOMMENDATIONS FOR FURTHER RESEARCH

This paper h a s utilized a m ethod of geometric program m ing to approxim ate the optim al service rate of two classes of queues: the

M /G /1 and M /M /s queues w ith linear waiting costs and server learning effects. It should be noted th a t no claim of global optim ality h as been m ade for th is algorithm, or, as of yet, for any geometric condensation algorithm.

This is som ew hat of a divergence from the cu rren t avenues of stu d y in geometric program m ing. Most recent work h a s centered on solution, via condensation of like term s, of unconstrained problems. Here a class of constrained problems, which proved solvable using condensation of unlike term s, was introduced. This m ethod m ay prove applicable to more unconstrained, single variable problem s with

polynomial denom inators (as the queuing problem s were before the su b stitu tio n and the resulting constraint).

This model and algorithm are applicable to the hypothetical shipping problem initially discussed; however, it is m uch more common

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th a t the m ean service rate be constrained (it was assum ed here th a t it w as not). A simple m ethod for approxim ation of the optim um of a bounded problem su c h as th is could be as follows:

1. Solve the unbounded problem with the geometric program m ing condensation m ethod of th is paper. 2. If the re su lta n t m ean service rate is not w ithin the

bounds th e n one of the bounds is the estim ated optim um.

3. If the resu lta n t m ean service rate is w ithin the bounds th e n it is th e estim ated optim um.

This will hold for the single server case b u t not for the m ulti-server case. T hat is, it m ay prove less costly to u se m ultiple servers if the m ean service rate is bounded. Analysis of w hen the m ulti-server queue is more economical is certainly worthy of more research.

A proof w as presented for single server optim ality of the M /M /s queue, which simplified the optimization m ethod significantly. Intuition m ight assert th a t the M /G /s is also subject to single server optimality. A straight forward expression for the expected num ber of custom ers in the M /G /s queue system w as not discovered in the research for this paper, b u t through stochastic m ethods sim ilar to those of Stidham

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(1968), single server optim ality m ay be provable. S uch proof would certainly extend the value of the GP condensation presented here.

The sensitivity of the optimal TEC to the service cost per custom er also deserves further investigation. Sensitivity analysis with regard to changes in the service cost per custom er and the waiting cost per custom er is possible with th is GP m ethod (referring back to eq. 3.1, k and col are dependent on the service cost per custom er and k, col, and co2 are dependent upon the waiting cost per customer), b u t intricate.

As a final note on queue design: th e literature search for this paper indicated th a t design is a nearly dead topic and all in terest is in queue control. As was, hopefully, dem onstrated with th is paper, there is m uch more to explore in th e field of queue design, be it other m ethods of

Figure

Figure  1:  Operation Costs versus  Service  Level
Table  4.1:  Example Problems  Used for Analysis Problem Mean ArrivalRate X Cost of Waiting cw Cost  of Servicec
Table 4.2:  R esults  o f Problem M /M /1A
Table 4.3:  R esults  of Problem M /M / IB
+7

References

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