TWO SERVICE UNITS WITH INTERFERENCE IN THE ACCESS TO SERVERS

We examine the service mechanism of two queueing models with two units in tandem. In the first model, customers who complete service in Unit 1 must wait in an intermediate buffer until the ongoing service in Unit II ends. In the second model, jobs can be pre-positioned in an intermediate buffer to await service in Unit II. Under the assumption of phase-type service times, the steady-state regime of the service system is studied in detail. The models are inspired by the gas pump model of A.B. Clarke and by phenomena observed in cafeteria lines and certain manufacturing systems. However, their primary interest may lie in the methodology of their exceptionally tractable analysis. We derive formulas for the throughput and other quantities by using the familiar PH-formalism. These formulas turn out to be unusually transparent and have probabilistic interpretations that do not depend on the PH assumptions. These interpretations therefore also hold for general service time distributions. The methodology is general and can be applied to other systems with interactions between servers. The models also present interesting algorithmic problems of didactic interest.


Introduction
We discuss two systems of queues with two service stations.Access to the second server or departures from the first station may be blocked by customers present in the other unit.In 1977 and 1978, in a series of preprints, A.B. Clarke introduced the first model.Informally described, it represents the interference one might see at two gasoline pumps with a narrow lane so that a customer who has completed service at the second pump may be blocked by a vehicle that is still using the first pump.
It was recognized that, under the assumption of exponential service times, the model called A.B. Clarke's Tandem Queue can be studied as a continuous-time Markov chain of GI/M/1 type.A fairly detailed analysis is found in Section 5.3 of Neuts [5].
Our model I is just Clarke's tandem queue with service time distributions of phase- type that are discussed in Chapter 2 of the cited book by Neuts.It is a fairly routine matter to extend the analysis of Markovian queueing systems with exponential assumptions to the corresponding models with PH service time distributions.One usually obtains Markov chains having the classical block structures but with blocks of large, often huge, dimensions.The resulting algorithmic problems are challenging but their discussion is not the purpose of this paper.
We shall show that the analysis with phase-type distributions can provide signifi- cant new physical insight that is lost in the multitude of special and interrelated simplifications due to the exponential assumptions.Important quantities can be given probabilistic interpretations that are independent of the PH-formalism.These interpretations are therefore also valid for general service time distributions.That points the way to the computation of these quantities by other analytic means or to their measurement through efficient simulations.
We concentrate on the throughput analysis of the two-server systems rather than on a full analysis of the queues.That is, we assume that an unlimited number of jobs are waiting for service and we analyze the departure rate of customers and va- rious other descriptors of the service mechanism.
The analysis of the models suggests interesting algorithmic problems of clear didac- tic interest.A numerical exploration will yield insight into queueing models with be- havior not seen in the classical models.

Description of the Models
In both models, there are two servers I and II with I placed to the left of II.The job flow is from left to right.Each customer is served by only one of the servers.All ser- vice times are independent random variable but the duration of those dispensed by Server I have the probability distribution F 1(. ); those of Server II obey the probabili- ty law F2(.).These probability distributions are of phase type.For j-1,2, F j(. has m, phases and the irreducible representation (/3(j),S(j)).The column vectors S(j) -S(j)e play their usual role in the formalism of PH-distributions.
The interest of Clarke's tandem model to queueing analysis lies in the quantifica- tion of the interference between the two servers.When there is a job in course in Unit II, a customer whose service is completed in cannot leave the system imme- diately.Server I remains blocked as long as his customer cannot move forward.To alleviate that blocking there is a finite waiting space between the two servers.The distinction between our two models lies in the way that this waiting space is used.
In distinguishing between the models, we see what happens at service completions in Unit I.In Model I, if upon such a completion Server II is free, the completed job leaves the system and two new jobs are initiated, one by each server.If Server II is not free, the completed job moves into a buffer of size n and Server I initiates a new job.Should that buffer be full, the completed job blocks Unit I.At the end of a job in II, that customer and all those that have already been served by I leave the sys- tem.Server II then remains idle until the job in I is finished.
In Model II, the buffer is used to pre-position customers for Server II.If upon a job completion in Unit I Server II is free, the completed job leaves the system and m + 1 new jobs are brought into the service system; one for each server to start processing and m-1 to wait in the buffer for service by II.These customers leave one at a time as they finish service in II.Jobs completed by I can leave the system only when there are not active jobs downstream.
With phase-type service times, the service systems of each of the models can be des- cribed as finite Markov chains, typically with many states, but with highly structured infinitesimal generators.To make that structure clear, we display these generators for small but representative values of n or m.By exploiting their special structure, we obtain explicit matrix formulas for various steady-state descriptors of the two models.These formulas can be used in numerical explorations from which substantial physical insight can be gained.Our approach is an example of the general modeling methodology advocated in [8]. 3. Discussion of Model I The state space of the Markov chain for Model I consists of n + 3 macro-states deno- ted by 0,1,...,n+2.The macro-state 0 corresponds to the case where Unit II is empty and it contains the m 1 labels of the phases for service time distribution FI(. ).The set of states n+2 contains the m 2 phases to service time distribution F2(.).It corresponds to the case where Server I is blocked.The macro-states 1,...,n-t-1 contain the mira2 phases of the two service times in course.Within each macro-state with 1 < < n the phases are listed in lexicographic order.The macro-state signifies that Server II is occupied and i-1 customers who have already been served by I are waiting in the buffer.For i-n-4-1, the buffer is full but the service in I is going on.
If that service terminates before that in Unit II, blocking occurs.The Markov chain then moves into the macro-state n+2.
We make extensive use of the Kronecker produce (R) and sum (R) of matrices.The properties of these operations are discussed in many books on matrix algebra; a summary of these for use with PH-distributions is given in Chapter 2 of [5].
The explicit matrix formulas for the steady-state probabilities involve two matrices M and N, each having a probabilistic significance to be explained later.
The matrix M is m 2 m 2 and is give by: The matrix N is m 2 m I and is given by: With irreducible representations of the two-phase-type distributions, the matrix M is irreducible and sub-stochastic.The matrix N is nonnegative and, as is easily verified, Me / Nee.Theorem 3.1: The vectors in the partitioned form of r are given by r(0) k*/(2 for l < k < n + l, and (n + 2) k*(2)M n + 1[_ S(2)]-1. (5) The normalizing constant k* is given by where ]21'(2) is the mean service time in Unit II.The quantity k* is the rate at which simultaneously services in both units are initiated.It is also the steady-state rate of departure of customers processed by Server II.
The quantity k* has several interesting interpretations.From its definition in (15), it is clearly the steady-state rate at which service in both units are simultaneous- ly started.The rate of terminations of jobs in Unit II is given by n+l which after substitutions and routine simplification reduces to k*.From the normalizing equation starting that r is a probability vector, we obtain [k*]-1 as the sum of three expressions.The first of these, r(0)e, is just the second term in the equation (6).
In the first expression, we write [-S(1) S( 2 Recalling the definition of M, routine matrix calculations now show that the sum of the two involved expressions reduces to (2)[-S(2)]-le-](2).
The Exponential Case Upon replacing S(1) and S(2) by the scalars -#1 and -#2 and /3(1) and /3(2) by one, we obtain the case of exponential service times discussed in [5].The infinite- simal generator Q1 agrees with the matrix A of formula (5.3.3)except that the latter corresponds to n 4.
The steady-state probabilities r(i) are as stated in formula (5.3.4) of [5].
The Departure Rate Next we calculate the steady-state rate #*(n) at which jobs leave the service system or the throughput of the system.The differential #*(n)dt may be interpreted as the expected number of completed jobs departing in a time interval of length dt.We ob- serve that, in steady-state, there are single departures at a rate r(0)S(1).Customers can also leave in groups of sizes i, 2 _< <_ n + 2, consisting of one customer processed by II and i-1 by Server I.
Theorem 3.2: The departure rule #*(n) is given by # The rate #*(n) is equal to A*(n), the steady-stale rate at which services are initia- ted.

Other Steady-State Descriptors
As for most other models analyzed by matrix-analytic methods, there are many use- ful descriptors of the steady-state regime of the service mechanism that are readily ex- pressed once we have the stationary probability vector r.The following is a selection of such descriptors.
The fraction p(n) of jobs served by Unit II, is given by the ratio p A typical idle time of Server II has the PH-distribution with representations [',oz(1)], where the vector , is given by 7 [(O)e]-lr(O), and its mean equals #'1(2idle, n)-[r(0)e]-lr(0)[--S(1)]-le.In addition, the mean time between an arbitrary visit to the macro-state 0 and the next is just [r(0)e]-1.
The fraction of time that Server II is idle is given by a descriptor of interest in quantifying the effect of the buffer size on the performance of the system.
The point processes of the departure epochs and of the epochs of initiations of ser- vices are both BMAPs.The BMAP is a Markovian point process for which many descriptors can be expressed in explicit matrix formulas.From its extensive litera- ture, we refer to Lucantoni [2,3], Neuts [7], and Narayana and Neuts [4] for discus- sions of the basic definitions and properties of BMAPs.We note that the order of the coefficient matrices of these BMAPs is the same as that of QI" The matrix formulas for most descriptors involve matrices of that order.
The principal challenge therefore lies in the efficient organization of numerical compu- tations with large but highly structured matrices.
From the present results, we immediately obtain expressions for various rates.
The rate of single initiations of service is = lr(i)[S(1)(R)e] while pairs of services start at rate k*.Jobs leave the system in groups of sizes ranging from 1 to n + 2. The steady-state departure rate #*(i; n) of a group of size is given by tt*(1; n)-(0)S( 1)+ ( 1)[e (R) S(2)], for2_<i_<n+l, and n) s( 2)] + + The sum #**(n) of these quantities is the rate at which departures occur regardless of the group size.By using formulas (15) and (4), we readily obtain that p**(n) k*[2 -/(2)M n + le].
The Matrices M and N These matrices arise naturally in studying the super-position of two independent PH-renewal processes; they were introduced in Latouche and Neuts [1].Consider the pro- duct Markov chain of the familiar Markov chains with generators S(1) + S(1)(1) and S(2)+ S(2)/(2), used in discussing the PH-renewal process.
The element M j, j, of M is the conditional probability that, given that we start at a renewal in the first process and with the second in phase j, the next renewal is also in the first process and when it occurs, the phase in the second process is j'.Similar ly, the element N j, h, of N is the conditional probability that, given that we start at a renewal in the first process and with the second in phase j, the next renewal is in the second process and when it occurs, the phase in the first process is h'.Most constants arising as descriptors of the present models can be interpreted as ex- pected values of random variables associated with the super-position of two indepen- dent PH-renewal processes.We given an illustration in the sequel.

Model I via Maxkov Renewal Theory
When, as in the case here, models based on phase-type or MAP analysis are unusually tractable, it is often possible to obtain slightly more general results by studying an embedded Markov renewal process.The succinct discussion that follows serves to illustrate that point.
We consider Model I at the ends of successive services in Unit II.We allow the service times in that unit to have a general probability distribution F2(.).The service times in Unit I have the PH-distribution with representation [fl(1),S( When a service in II ends while Server I is occupied, we include the phase of that service in the state description.The rn 1 corresponding states form the macro-state 0. The single state 1 signifies a transition epoch in which new services simultaneously start in both units.It is readily seen that we obtain a Markov renewal process with m 1+ 1 states.We write its transition probability matrix K(.) in the partitioned form: The elements of K(x) are obtained by routine conditioning arguments.To write them concisely, we need the standard matrices P(u, t) for the counting process of the PH-renewal process with the underlying distribution FI(. ).To remind us that these correspond to the service time distribution in Unit I, we add the subscript 1.
The matrix K(0, 0, x) is given by x y K(O, O, x) exp[S(1)ulS( 1)du(1) E P1 (r; y tt)dr2(Y u), 0 0 r=O for x _> O.To obtain that expression, we condition on the end of the service in course in Unit I and on the number of additional service time terminations in I during the service by II.There can be at most n such job completions; otherwise, blocking occurs.
There are similar expressions for the other elements of K(x), but for the sake of brevity, we only give the expressions for their Laplace-Stieltjes transforms which facili- phases for the service time distribution FI(. ).The macro-states 1,...,m contain the m lm2 phases of the two service times in course.The corresponding macro-states with an asterisk contain the phases of the service in course in Unit II.Server I has then completed a job but is blocked so we need not keep track of a service phase.
The infinitesimal generator Q2 for Model II is displayed for m 3. The structure for a general value of m is similar.As for Model I, two matrices M 1 and N 1 appear in the explicit matrix formulas for the steady-state probabilities.These matrices are defined as follows: The matrix The matrix N is m X m 2 and N [I (R)/3(2)][-S(1)(R) 5'(2)]-l[s(1) (R) I]. (31) The matrix M 1 is irreducible and sub-stochastic.The matrix N 1 is nonnegative and Mle + Nle e.
Theorem 4.1: For Model II, the vectors in the partitioned form of r are given by and r(0 for 1 <_ <_ m.The normalizing constant h* is given by [h*] m (2) -3(1)M[ S(1)]-le, (35)   where #1(2) is the mean service time in Unit II.
Proof."The required calculations are similar to those for Model I. We present only service in II also releases a customer blocked in Server I. (43) In Neuts [5] it is shown that, with exponential servers of unequal rates, the throughput in Model I is largest with the faster server is assigned to Unit I.For Model II and with the same m, the opposite conclusion holds.
Given the complex dependence on the service time distributions it is by no means obvious that the same conclusion holds for non-exponential distributions.For PHdistributions, a numerical exploration of that question is easy.
A further interesting question is to find the value of m for which the throughput is largest.For the exponential case, that is elementary; for fixed , we find the m for which the function in (43) attains its maximum.For PH-distributions, the numeri- cal exploration is more substantial.

Some General Methodological Comments
While the two service systems in this paper may be of limited practical applicability, their analysis is similar to that required for other tandem queues and for the complex systems arising in manufacturing and telecommunications.
We have demonstrated how the formulation with PH-distributions can provide structural insights that are lost under exponential assumptions.That formulation also makes clear how much or more commonly, how little-analytic tractability of the model is to be expected.The models in this paper are highly tractable.For others that are less, the computational effort required to study large, structured Markov chains becomes clear.Such an effort is worthwhile; it can be methodological- ly challenging and its results provide benchmarks for the more common simulation