Parallel queuing systems with and without decentralized server reallocation — A simulation study

Parallel queuing systems with and without decentralized server reallocationA simulation study single ownership). Examples o f the former are transportation firms operating in the same mode but in different geographical markets, and of the latter d o w n t o w n and airport offices o f vehicle rental companies. Due to a variety of constraining factors, some institutional and others economic in nature, operational interaction among these organizational entities is often not actively pursued. 'Interaction' in this instance refers to the temporary transfer of resources from one location to another to cope with anticipated overcapacity in one and undercapacity in the other. Standardization, discreteness and transferability o f the resource itself and/or its productive capacity are intuitively essential prerequisites to this kind of resource sharing. Besides transportation equipment, resources that generally satisfy these requirements include tools, materials handling, office and instructional equipment as well as labor (at least within the same organization). In the few cases where operational interaction is practiced it takes place under centralized control i.,e., all transfer decisions are made by a single (global) manager, Initial findings of a comprehensive research program described herein suggest that decentralized (local) control is a viable alternative worth further investigation. The rationale behind this research effort is not found in the belief that decentralized resource reallocation is inherently superior to centralized reaUocation. On the contrary, mathematical programming models - at least from a global perspective should result in better performance assuming of course deterministic or at least stationary stochastic demand, no decomposition difficulties, no problem size limitations, etc. However, there are situations where not only is stochastic demand at each location non-stationary but furthermore this non-stationarity can not be adequately described beforehand. Alter. natively there may be extraneous considerations such as information overload at the global level plus unavailability or high operational cost of a mathematical programming model - which might make centralized decisionmaking undesirable. In cases such as these local planning might be more attractive

L e o n i d a s C. C H A R A L A M B I D E S The Robert A. Johnston College o f Bushless Administration, Marquette University, Milwaukee. WI, U.S.A. Received May 1979 Revised October 1979 This paper describes tile results of an initial exploratory .esearch effort on the subject of the dynamic reallo.cation of servers in a group of parallel queuing systems under decentralized control. Each queuing System is described as being (M/Ek/ci, r) : (I"CI:S[~*/~*) where t = 1..... ~ and i = 1..... I. Server reallocation is handled by a (G(b)/G/C) : (FCS/C]K) system - where C = Z ~ l X~/=l Ci,r, and K is an unknown constant - located at a higher level than the group members. The customers of the latter system are requests for servers independently generated by identical heuristic event-triggered operating policies at each of the lower level systems. These local policies are of the stochastic review, relative control number type. Performance is measured by means of a variety of criteria from the viewpoint of the g~oup as a whole (global) as well as an individual queuing system (local). Results of experiments using a simulation model confirmed the null hypothe,.is that the customer service bchaviour of a group that shared servers was significantly different from the behaviour of a control group that did not share servers.

I. Introduction In a number o f organizational settings users o f intermittently employed resources are geographically dispersed to an extent that prevents the consolidation of separate stocks of these resources into a single location. Thus, each stock ha~ its own management and serves a distinct demand concentration. It may either be an independent goal seeking organization (implying multiple ownership o f the total stock) or a distinct entity of a larger organization (implying The author wishes to express his appreciation to his colleague, Professor David Allen of the Department of Management, Marquette University for his invaluable experimental design suggestions and computer simulation model validation assistance. © North-Holland Publishing Company European Journal of Operational Research 6 (1981 ) 46 -55 46

L. C CharalamhMes / Parallel queubzg systems

because of the alleged superiority of portions (i.e., forecasted) of the informational input. More specifically if (a) the resources are relatively homogeneous, (b) the number of prospective customers at each location is reasonably small and (c) the local administrator is sufficiently close to them 'informationally' (for example where the former and the latter belong to the same organization), then the subjective estimation of the pertinent demand probability density function by the local administrator may be more accurate than a formal (and most likely centralized) forecasting method if that individual is also authorized to be a decisionmaker. That is, if he is allowed to defend his intuitive forecasting process with the assumption of risk on his part concerning initiation and even consummation (through some form of bidding or negotiation process) of inbound and outbound resource transfers. Whether or not such a management approach will succeed in a specific application, at least from a global viewpoint, will depend in part upon the local manager compensation incentives utilized as well as the risk propensity profiles (utility functions) of the participating individuals. However it is reasonably safe to expect that a number of these local managers, and consequently the organizational entities under their command, will be better off than if these persons were just passive conveyors of locally generated historical demand data. In summation therefore, the justification for this research is founded in the conviction that given suitable parametric conditions (a) resource reallocation under local forecasting and decisionmaking combined with (b) competitive assignment of scarce resources, can be a more attractive alternative, globally and/or locally, than (i) resource reallocation under global forecasting and decionmaking followed by (ii) some kind of global to local adjustment mechanism. However, to avoid raising any validity questions concerning arbitrary comparisons between centralization and decentralization this research uses no reallocation as the benchmark case. Thus it hypothesizes that in certain situations a group of organizational entities which choose to interact - under conditions of nearly coml;d "te local control - can achieve productivity that is superior to that realized by non interacting entiti,,s collectively and/or individually. Furthermore

47

inasmuch as this is the initial effort on the subject (a) forecasting is not modelled and (b) scarce resources are rationed through the simplest priority assignment scheme possible (FCFS~. Thus attention is simply focused on finding whether local elementary decisionmaking policies functioning in a perfectly random environment can have a signi~cant effect upon selected dimensions of global and/~r local performance.

2. Related literature This paper studies the multiple objective function parallel multifaci!ity version of the variable capacity queuing system problem. It draws from two related management science areas: dynamic formulations in queuing theory of multichannel single-phase systems as well as inventory theory of nonconsumable items. In the former case the decisionmaker manipulates the utilization pattern of the resource (servers) while in the latter he is responsible for the input rate of the resource (inventory) into the service system. Controlling the output rate of the resource from the system hasgenerally not been the concern of dynamic inventory control optimization models. Notable exceptions are the works of Fukuda [5] and Whisler [20]. Ill variable capacity queuing systems, on the other hand, the decisionmaker controls the input as well as output rate ef servers from and to the environment respectively. Early research papers in the area (see Rutenberg [15], Moder and Phillips, Jr. [11], and Yadin and Naor [21]) sought to develop single system operating policies which are functions of expected values of certain performance criteria like queue length or queuing time. Recent attention has been directed toward finding cost based optimal control policies the structure of which allow server 'holding' and customer waiting costs, as well as an instantaneous switching cost incurred whenever the number of servers in the queuing system is changed (see Bell [1,21). The single objective function, parallel multifacility version of the variable capacity queuing system problem coincides with inventory stock redistribution models and, to some extent, certain job shop scheduling models. Representative examples of dual resource constrained job shops relevant to this study are found in the works of Holstein and Berry [10], Fryer [6] and Gunther [7] which consist of a search for efficient labor transfer eligibility and labor transfer assiglmlent

48

L. C Charalambides / Parallel queuing systems

decision rtlles. The former class of rules is used to decide when a laborer should be considered for reassignment while the latter class is used to determine the destination of the laborer. In all cases servers are transferred directly from one queuing system to another rather than through a neutral entity as in this research. The nature of the decision rules employed in most of these models implies a significant informational input on the status of all the facilities in the job shop, rather than primarily the originating facility as in this effort. This is the case for instance with all three versions of the assignmnet rules studied by Weeks and Fryer [19]. On the other hand one of the t~4o versions of the eligibility rules in their work - the one allowing for complete decentralization - considers only local information. However, by its inherent nature an eligibility rule alone cannot be used to determine the feasibility of completing a potential server transfer. Consequently these rules implicitly presume the existence of a single global objective function.

3. The model A simulation model defined within a queuing theory framework is employed. Monte-Carlo simula-

tion has not been used in previous studies of variable capacity queuing systems since neither the postulated structure of the models nor the nature of the operating policies studied have warranted it. Its use in this case is deemed appropriate because the complexity and dynamic nature of the interaction discipline observed by the group of queuing systems introduces an element of uncertainty that cannot be deductively treated. The main modeling vehicle is the single line, single phase, multiple channel, variable capacity queuing system. A number of these queuing systems comprises the Service System (Fig. 1). It is composed of two echelons: a lower - i.e., closer to the customers comprised of a number of Customer Queuing Systems (C.Q.S.'s) and an upper comprised of a single Request Queuing System (R.Q.S.). In the lower echelon a parallel structure is postulated i.e., each C.Q.S. has its own unique Customer Population. The Source Population is the composite of these Customer Populations. The difference between a Customer and a Request Queuing System is that the first serves a Customer Population, an entity external to the Service System, while the second serves the requests for servers from the C.Q.S.'s which are entities internal to the Service System. Consequently, the Service System can not be

LOWER ECHELON

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F~g. 1. The Service System.

B .tA~E S~E.

49

L. C. Charalambides / Parallel queuing systems

viewed as a queuing network (see Disney [4]). The transportation times between each of the C.Q.S.'s and the R.Q,S. are postulated to be deterministic and the same for all C.Q.S. and R.Q.S. combinations (both directions). Each C.Q,S. is a separate organizational entity composed of a queue of waiting customers (the Customer Queue) and a Customer Service Facility of available (i.e., the Base) and unavailable servers (Fig. 2). There are no limits on either customer patience or Customer Queue length. All C.Q.S.'s practice a simple First Come-First Served (FCFS) service choice method from among waiting customers• The Customer Populations utilize the stationary Poisson arrival rate distribution which is generally in agreement with related research on inventory control theory for spares and repairables (see for example Hadley and Whitin [8], Simon [17],Simpson [18] and others) as well as variable capacity queuing systems (for instance Heyman [9] and Schleef [ 16] ). Although conveniently used as a service time distribution in a variety of queuing formulations (see Rosenshine [14]), the negative exponential is not employed here because it is felt that is is unduly restrictive. Parikh [13] found that service times distributed according to the Er]ang-k with values of k in the range of 6 - 1 2 do a satisfactory job in modeling real world distributions of such recurrently used assets as meter vehicles. Thus, an Erlang-k with a value of K = 9 is chosen as a common customer service time distribution for all C.Q.S.'s. The number of C.Q.S.'s is a parameter significant enough to warrant investigation as an experimental factor e.g., a small configuration (four), and a large

configuration (eight) are examined. The finite number of servers shared by the C.Q.S.'s is the same in the small as well as the large configuration i.e., tile 'supply' of servers is fixed. Another experimental factor deals with the nature of tile Source Population parameters: a homogeneous Service System is defined as one in which the types, means and variances of all the Customer Population arrival rate distributions are the same. In a heterogeneous Service System, on the other hand, some or all of those parameters are nonuniform• For example the Customer Population arrival rate distributions may be of the same type and have the same mean, but their variances may be different. In this effort the attribute that describes the degree of homogeneity is the arrival rate of each queuing system. Thus in a homogeneous Service System all C.Q.S's have the same arrival rate mean while in a heterogeneous Service System three fourths have the same value and one fourth have a different one. The various simulation model parameter values are a priori determined using a heuristic, computer based numerical search procedure which is totally independent of the model. These values are the total number of Service System servers~ the mean arrival rate of the Source Population, the (fixed) mean service rate per C.Q.S., the initial server allocations to the C.Q.S.'s and the distributional forms of the N Customer Population arrival rate distribution means and variances. This procedure, which requires analyst participation, ensures that the mean of the Source Population arrival rate distribution is uniform for all configurations e.g., small, large, homogeneous and heterogeneous.

J

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Fig. 2. The Customer Queuing System.

50

L. C Charalambides / Parallel queuing systems

The generic control-type operating policy used at each C.Q.S. is composed of four decision rules which pertain to the timing and extent of the increase or decrease of the number of servers currently residing at a C~Q.S. It is heuristic in nature and revolved around a modification of the concept of insta~taneous server utilization at a C.Q.S. That is, the numerator of the so called utilization factor (u.f.) is the number of servers in use at a C.Q.S. plus the number of customers in the Customer Queue. The denominator, on the other hand, includes the total number of servers at that C.Q.S. (busy and idle) plus the number of servers in transit to that C.Q.S. plus the number of outstanding server requests from that C.Q.S. at the R.Q.S. Queue. The upper and lower control limits (0.85 and 0.65 respectively) against which the u.f. is occasionaly compared are neither a function of time nor a function of the status of the Service System or any of its parts. They are established using preliminary simulation runs and the results of the heuristic numerical search procedure mentioned above. The occasions on which the u.f. is computed and thus server transfer transactions triggered are the three independent events in the operation of the Service System i.e., the events that are a direct function of the stochastic processes driving the System. They are: (a) the arrival of a customer at a C.Q.S., (b) the release of a server by a customer and (c) the arrival at a C.Q.S. of a requested server from the R.Q.S. Thus if the u.f. is found to be greater (less) than 0.85 (0.65) a request for (a dispatch of) servers is initiated sufficiently large to bring the u.f. to 0.75. If the u.f. lies between the two limits then of course no action is taken. The computer program that describes the model was written in the General Purpose Discrete Simulator (GPDS) language. This is the XEROX version of the well known and widely used IBM General Purpose Simulation System (GPSS) language. FORTRAN subroutines were employed for u.f. computations and data gathering chores.

4. The experimental framework This research tests the following two broad categories of null hypotheses: (a) Global performance of a non-interacting group

(a SET) of entities is not significantly different from the global performance of an interacting group (a SYSTEM), and (b) Local performance of a set of entities is not significantly different from the local performance of a system of entities. Three experimental factors - each at two levels are employed: Factor A. Presence or absence of operational interaction among the C.Q.S.'s (the interaction factor). LEVEL-l:

No interaction,

LEVEL +1:

Interaction.

Factor B..Number of C.Q.S.'s in the Service System (the group size factor):

LEVEL-l:

Small configuration (four),

LEVEL +1:

Large confguration (eight).

Factor C. Composition of the Source Population i.e., the form of the relative frequency distribution of the Customer Population arrival rate means (the group composition factor).

LEVEL - 1 :

Homogeneous configuration,

LEVEL +1 :

Heterogeneous configuration°

Since there are just three factors a full factorial of 23 = 8 simulation runs only needs to be executed. The steady state period of each simulation run (experiment) is divided into thirty subruns (replications). In an inductive study such as this the total number of experimental factors chosen as well as the number Of levels for each is constrained by modeling considerations peculiar to the problem and computer resource availability. However, the choice of the particular sample of factors from the population of all possible factors reflects the researcher's a priori estimate of their relative importance to the key performance criteria. The limited number of secondary - to interaction factors is explained by the fact that the findings reported herein are part of a broader research program, one of the objectives of which was the study of the desirable characteristics of the operating policy. That objective necessitated the intro. duction of four more factors associated with selected operating policy characteristics plus another representing interechelon transportation time. Inasmuch as this research constitutes the initial effort on the subject the introduction of a cost struc-

L. C Charalambides / Parallel queuing systems

ture is considered premature. Consequently. a number of diverse performance criteria have to be employed to evaluate the performance of the group under the various experimental conditions. Criteria observations are recorded for each subrun. Corresponding to each global (echelon wide) criterion is a group of eight local (C.Q.S.) criteria. Global and local criteria are divided into two categories: the customer associated criteria category includes criteria that provide a measure of the 'quality of service' accorded the customers while the server associated criteria describe the level of activity of the servers. Analysis of variance is conducted through the use of a regression model. In symbolic notation the general form of this model is as follows: J )'i = ~

1=o

b/xi, ] + e i ,

where i is the identification number of a replication; /" is the identification number of an independent variable i.e., an individual experimental factor or statistical interaction among factors. This index is allowed to take on a lower value of 0 so that the mean effect ,:.an also be modeled. Higher than third order factor interactions are assumed insignificant and thus not included in the model; Yi is replicate i of the dependent variable i.e., a specific global or local performance criterion;

5i

bj is the regression coefficient (also called 'effect') of independent variable j; x i , / i s the level of independent variable ] in replication i; e i is the error terx~ associated with replicate i. The assumptions required by regression analysis for inference-making purposes appear to be generally satisfied. The usual parametric tests of hypotheses and confidence intervals used in regression analysis are 'robust' as regards to mild deviations from the stated assumptions. However in this study, as in similar simulations of large scale systems, the homoskedasticity assumption was found to be consistently violated. Therefore, marginally significant regression coefficients are not used as supporting evidence in testing the null hypothesis that a specific regression coefficient is equal to zero given that a!l the independent variables are in the model.

5. Research findings Discussion of the regression results resolves around the identification of the regression coefficients that are significant above the five percent level, the signs of these significant coefficients, and the relative order of magnitude of the significant F ratio values. The sign of a coefficient provides more information than

Table 1 Regression coefficients (B) and F-ratio values (F) of Independent Variables (I.V.) for Global Performance Criteria, Mean Lower Echelon Queue Length (I), Probability of Zero Lower Echelon Queue Residence Time (II) and Mean Lower Echelon Queue Residence Time (lid I

I1

III

I.V.

B

F

B

F

B

F

A B C

-0.204 a) 0.435 a) 0.082 -0.258 a) 0.040 0.002 -0.038

14.123 64.051 2.298 22.603 0.545 0.002 0.497

0.013 a) -0.020 a) 0.000 0.012 a) 0.001 -0.002 0.000

31.117 75.235 0.060 29.704 0.396 1.097 0.012

-12.142 a) 25.558 a) 4.733 -15.325 a) 2.350 - 0.083 - 2.400

15.516 68.753 2.358 24.719 0.581 0.001 0.606

AB

BC AC

ABC Mean R-sq F-~

0.672 0.3098

a) Significant above the 0.05 level.

0.961 0.3723 14.874

38.908 0.3266 19.660

16.076

L. C. Charalambides / Parallel queuing systems

52

Table 2 Regression coefficients (B) and F-ratio values (F) of Independent Variables (I.¥3 for Global Performance Criteria, Mean of the Traffic Intensity Distribution of the Lower Echelon (IV), and Mean Number of Servers in the Lower Echelon (V) IV

V

I.V.

B

F

B

F

A B C AB BC AC ABC

1.010 a) 0.111 -0.212 -0.145 -0.038 0.053 0.027

31.878 0.388 1.400 0.659 0.044 0.088 0.022

-3.615 a) 0.395 -0.183 0.395 -0.121 -0.182 -0.121

190.311 2.277 0.486 2.277 0.214 0.486 0.214

Mean R-sq F-gr

75.755 0.1294

236.385 0.4583 4.q26

28.038

a) Significant above the 0.05 level.

the other characteristics because it delineates the slope of the response function for that criterion as one moves from one factor level of an independent variable to the other. The desirable optimization type for each specific criterion is estabhshed prior to its analysis. Consequently, the sign of the coefficient points to the direction in which the independent variable should gravitate.

5.1. The first category of null hypotheses Tables 1 and 2 include F ratio values (represented by the symbol F) and partial regression coefficients (B) of all the independent variables for five global criteria. They also contain the mean effect (Mean), the R-square or coefficient of determination (R-sq) as well as the value of the overall F statistic for the entire regression equation (F-gr).

5.1.1. Customer associated criteria Undoubtedly the most salient finding related to Table 1 is the fact that for all three performance criteria the operational interaction level of factor A is preferable to the no interaction level. However, it is not the strongest of the three experimental factc~rs: the magnitudes of the F ratio values of this factor for each of these criteria are at most about half that of the F ratio values of factor B. The joint effects (AB)

of these two factors are also significant. Their signs though do not support the respective choices of the factor levels indicated by the main effects analysis. Furthermore, their net effect is not advantageous from a systems design viewpoint i.e., if factor B adopts level +1 (large configuration) and factor A level +1 (operational interaction) the F ratio value of B outweighs the sum of the F ratio values of A and AB for all three criteria. This means that the cumulative advantages of interaction can be wiped out by the disadvantages of large Service System size. Since the desired optimization type ef Criteria I and Ill is the opposite of that for II, it is not surprising to find that the signs of all significant coefficients are reversed between these two sets of criteria. Even though the group composition factor appears to be more important for Criteria I and Ill than for Criterion II it is relatively less important than the operational interaction factor. A comparison of the signs of the three factor joint effects (ABC) upon these three criteria with the corresponding two factor joint effects lends further support to this conjecture. That is, every joint effect that includes factor C is significant. It should be noted that, even though the coefficient of determination values are in the thirties, all three regressions are significant beyond the five percent level. Due to the very short average waiting time (in the hundreds of time units) relative to the average service tinge (in the thousands), the R-square values of the regressions of Mean Number of Customers in the Lower Echelon and Mean Lower Echelon Residence Time criteria are very small and thus not shown in a separate table. Only factor B has any significant effect and that is for the latter criterion. However, the signs of the effects of factor A on both criteria are in agreement with those for Criteria I and Ill in Table 1. The same holds true for the two factor interactions between factors A and C. Generally the dominance of the group size factor for service associated criteria is to be expected on an intuitive basis of course i.e., the larger the number of C.Q.S.'s the more likely it is for a server to be at the 'wrong' C.Q.S. at a particular point in time thus resulting in higher lower echelon queue lengths, longer echelon queue residence times, etc.

5.1.2. Server associated criteria The signs of the two significant effects of factor A and the magnitudes of the respective F ratio values

L. C. CharalambMes / Parallelqueu¢.ngsystems in Table 2 indicate that there is a definite increase in the utilization of a system as compared to a set of C.Q.S.'s. The other two factors do not have any significant effects, whetl,er main or joint. Both of these findings are not surprising since when operational interaction is in effect servers may reside in locations other than the group, namely in transit or at the R.Q.S. Thus they spend less time at the group, in a way justifying the positive effect upon the traffic intensity and the negative effect upon the number of servers to be found at the group. Both regressions are significant although the large value of the mean effects relative to the respective independent variable effects suggests that a rather small period of time was spent by the servers away from the group.

5.2. The second category o f null hypotheses The results of the regressions for the local criteria generally support the results for the global criteria above. The presentation scheme employed for Tables 3 and 4 includes the following informaP.on for each of the three experimental factor independent variables: (a) the range of values of the significant, above the 0.0064 level l, regression coefficients among the eight C.Q.S. associated coefficients (Range B), (b) the value of the smallest F-ratio associated with the significant coefficients (Min F) and (c) the number (N) of significant coefficients. The same information is given for any two or three way interaction independent variable that has at least one significant C.Q.S. regression coefficient 2.

1 The reason for choosing the 0.0064 leve! is so that the results of both categories of null hypotheses are reported at approximately the five percent level. According to Ott [ 121 the 'overall error rate' (~e) for a number of comparisons (c) is given by ~e = 1 - (1 - a)c where o~is the Type I error of an individual test. For our purposes of course ae = 0.05 and c = 8 resulting in an o~of 0.0064 percent. 2 Because of a unique modeling predicamenI with this experimental framewozk Tables 3 and 4 incorporate the results of two kinds .of regression analyses i.e., one with factor B and the other without. This is due to the manner in which these two factors were modeled: to sim ~late a small heterogeneous configuration C.Q.S.'s 1 to 3 were the 'same' (with respect to interarrival time distribution means) while 4 was the 'different' one. For a large heterogeneous configuration, on the other hand, C.Q.S.'s 1 to 6 were the

53

5.2.1. Customer associated criteria As table 3 indicates only a small number of factor A effects is significant for Criteria I, I1 and 111. This can be attributed to the strength of factor B relative to the others which of course confirms the findings pertaining to the corresponding global criteria in Table 1. However, what is significant, is that the signs of all these effects are in agreement with those determined in the analysis of the corresponding global criteria i.e., the operational interaction level is more desirable than the no interaction level for individual C.Q.S.'s. Thus, even though the frequency of significant regression coefficients at the postulated level of significance is low, the sign of every local regression coefficient of factor A is the same for each family of local performance criteria. Obviously if the level of significance were to be relaxed, the statement could be made that, at least as far as quality of customer service is concerned, interaction is an attractive proposition to a sizable number of C.Q.S.'s.

5.2.2. Server associated criteria The main difference between Tables 4 and 2 is the appearance of a number of significant effects - in the case of criterion V - attributed to factor B as well as the 'sign split' of the coefficients for factor C. Both discrepancies are attributed to the modeling problem with factor B mentioned above. Of the eight significant coefficients for factor C five are negative and three positive. The former are associated with C.Q.S.'s that were always the 'same' while the latter with C.Q.S.'s that were "different'. A tentative conclusion that can be inferred therefore is that local C.Q.S. performance is to some degree affected by local conditions.

6. Conclusions The answer to the research question posed in this effort is that from a global criteria viewpoint there

same while 7 and 8 were different. That is, to represent level -1 of factor B, C.Q.S.'s 5 to 8 had to be deactivated in half of the simulation runs. Consequently under these circumstances the values of the relevant performance criteria for C.Q.S.'s 5 to 8 were a priori known to bc equal to zero.

54

L . C. C h a r a l a m b i d e s

/ Parallel queuing

systems

Table 3 Number (N) and Range (Range B) of Significant Regression Coefficients as well as the Minimum F-ratio Value above the 0.0064 Significance Level (Min F) of Independent Variables (I.V.) for Local Performance Criteria (I), (II) and (lid Mean Queue Length at a C.Q.S. ~I) I.V.

Range B

Min F

A B C

. . . . . . . . . 0.051 0.062 8.628 -0.057 9.039

AB

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Probability of Zero C.Q.S. Queue Resi- Mean C.Q.S. Queue Residence Time Oil) dence Time (1I)

N

P,ange B

0 0.020 2 -0.034 1 0 0.018

0.037 -0.025 0.023 0.019

Min F

N

Range B

10.225 22.964 15.334 8.676

2 . . . 4 28.667 1 2

.

.

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.

. . 47.058

-26.775 -28.292

Min F

N

14.084 10.279 7.617

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.

0

Number of Significant Values of the F-ratio due to Regression: (I): 3, (1I): 7, (III): 5. Range of all R-square Values: (1): 0.02770.1098, (1i): 0.083-0.1875, (III): ~3.0582-0.1697. appears to be a significant difference between the performance of an interacting group of variable capacity queuing systems and the performance of a non interacting group of fixed capacity queuing systems. That is, the former is generally able to provide superior customer servi~:e than the latter. As expected t ~ s superiority is marginal; the randomness of two of the underlying stochastic processes is a formidable barrier to effective operating policy design. Nevertheless the finding is noteworthy inasmuch as an oversimplified Service System was postulated and a rudimentary generic operating policy was utilized. This of course was necessitated by the desire to follow a strictly evolutionary research path. A combination of a richer simulation model and a better policy, 'better' in terms of control sensitivity,

would likely result in stronger main effects of the operational interaction experimental factor upon the same criteria as the ones examined in this effort. This would be especially true if nonstationary arrival and/or service stochastic processes were assumed. In that case adaptive control polices should obviously be examined. It should be pointed out that the question of superiority o f interaction over non interaction in an economic sense was not addressed. Thus, to determine if interaction can 'pay' for itself, a value should be placed not only on the time o f the customer but also on the process of affecting a server transfer. In this research only partial data concerning server trans. fe~s could be collected and hence was not analyzed i.e., obviously no transfers can take place when the

Table 4 Number (Ar) and Range (Range B) of Significant Regression Coefficients as well as the Minimum F-ratio Value above the 0.0064 Significance Level (Min F~ of Independent Variables (I.V.) for Local Performance Criteria (IV) and (V) Mean of the Traffic Intensity Distribution of a C.Q.S. (IV) I.V.

Range B

Min F

N

8.535

3 0 0

Mean Number of Servers in a C.Q.S. (V)

Min F

N

- 0.789 - 14.125 147.694

26.437 6732.871 786.563

2 4 8

A B C

1.042 1.933 . . . . . . . . . . . . . . . . . .

AB

.

BC

. . . . . . . . .

0

AC

.

.

.

.

.

.

.

.

.

0

.

.

.

.

.

.

.

.

.

0

ABC

.

.

.

.

.

.

.

.

.

0

.

.

.

.

.

.

.

.

.

0

.

.

.

.

.

.

.

.

0

- 0,900 --26.618 -49.216

Range B

.

.

.

.

.

.

- 11.580

.

.

.

73.600

0

I 1.904

Number of Significant Values of the F-ratio due to Regression: (IV): 2, (V): 8. Range of all R-squares Values: (IV)" 0.031330.1080, (V): 0.8720-0,9916.

4

L.C. Charat~ambides/ Parallel queuing systems model operates in the no interaction mode (e.g., as a set). The findings with respect to the corresponding subsets of local criteria are at best spotty and thus inconclusive - at least at the five percent overall level o f significance. However, the results that do pass the test are in almost complete agreement with the global criteria as far as the identity o f the desirable factor levels is concerned. The tentative conclusion therefore is that what is 'good' for the group is to an extent 'good' for the individual queuing systems. This suggests that bringing together independent queuing systems under a scheme of 'cooperative competition', similar to the one described herein, should have appeal to a certain number of them. Whether this number will constitute a clear majority o f such participating systems should be further investigated under progressively more realistic experimental conditions. These conditions may pertain to: (a) group composition variability with respect to service times, (b) interaction transportation times (deterministic or stochastic, the same for each C.Q.S. to R.Q.S. pair or different, etc.), (c) finite Customer Populations, (d) various degrees and forms of customer patience, (e) multiple phase arrangements, (f) a finite number of non-shared servers, and (g) operating policies that differ from one C.Q.S. to another (e.g., complete decentralization). The second condition is particularly important since this research postulated instantaneous resource transfer between a C.Q.S. and the R.Q.S. - something encountered in time-sharing computer systems only.

References [ 1 ] C.E. Bell, Turning off a server with customers present: Is this any way to run an MIMIc queue with removable servers?, Operations Res. 23 ( 1975) 571 - 574. [21 C.E. Bell, Optimal operation of an M/M/2 queue with removable servers, Working paper No. 37, College of Business Administration, University of Tennessee (1976).

55

[3] L.C. Charalambides, A simulation study of the decentralized reallocation of servers in a group of parallel variable capacity queuing systems, Unpublished Ph.D. Dissertation, College of Business Administration, University of South Carolina, Columbia, SC (1978). [41 R. Disney, Random flow in queuing networks: A review and critique, AIIE Trans. 7 (1975) 268--288. [51 Y. Fukuda, Optimal disposal policies, Naval Res. Logist. Quart. 8 (1961) 221-227. [61 J.S. Fryer, Effects of shop size and labor flexibility in labor and machine limited production systems, Manag. Sci. 21 (1975) 507-515. [7] R. Gunther, Dual resource parallel queuing systems with server transfer delays, Paper presented at the Joint National Meeting of ORSA/TIMS, Atlanta, GA, 1977. [8] G. Hadley and T.M. Whitin, A model for procurement, 'allocation and redistribution for low demand items, Nav,d Res. Logist. Quart. 8 ( 1961 ) 395 -414. [91 D.P. Heyman, Optimal operating polices for M/G/I queuing systems, Operations Res. 16 (1968) 362-382. [lOl W.K. Holstein and W.L. Berry, The labor assignment decision: An application of work flow structure information, Manag. Sci. 18 (1972) 390-400. [111 J.J. Moder and C.R. Phillips, Jr., Queuing with fixed and variable channels, Operations Res. 10 (1962) 218-231. {121 L. Ott, An Introduction to Statistical Methods and Data Analysis (Duxbuty, North Scituate, MA, 1977). [131 S.C. Parikh, On a fleet sizing and allocation problem, Manag. Sci. 23 (1977) 972-977. 1141 M. Rosenshine, Queuing theory: The state of the art, AIIE Trans. 7 (1975) 257-267. [151 Y.H. Rutenberg, Sequential decision models, US Government Research Report AD256- 217 ( 1961 ). [161 H.J. Schleef, The MIMIc queue with removable servers: 'Fheory and computation, Paper presented at the Joint National Meeting of ORSA/TIMS, Atlanta, GA, 1977. [17] R.M. Simon, Stationary properties of a two-echelon inventory model for low demand items, Operations Res. 19 (1971) 761-773. [181 V. Simpson, Inventory theory lor repairables, Ph.D. Dissertation; School of Engineering and Science, New York University, New York (1972). 119] J.K. Weeks and J.S. Fryer, A simulation study of operating polices in a hypothetical dual-constrained job shop, Manag. Sci. 22 (1976) 1362-1371. [20] W. Whisler, An inventory model for rented equipment. Ph.D. Dissertation, School of Engineering, University of California, Berkeley, CA (1965). [211 M. Yadin and P. Naor, On queuing systems with variable service capacities, Naval Res. Logist. Quart. 14 t 1967) 43-53.