Modelling M / G / 1 queueing systems with server vacations using stochastic Petri nets

The theory of non-Markovian stochastic Petri nets is employed in this paper to derive an alternative method for studying the steady state behaviour of the M/G/1 vacation queueing system with a limited service discipline. Three types of vacation schemes are considered, and sytems with both a finite population and those with an infinite population (but finite capacity) are considered. Simple numerical examples are also provided to illustrate the functionality of the methods and some useful performance measures for the system are obtained.


Introduction
Queueing systems in which the server is sometimes inactive (taking vacations or working elsewhere) while customers are waiting for service may find many applications in the performance modelling of computer and communication systems.For example, in many digital systems, the processor is multiplexed among a number of jobs and is hence not available all the time for a single job type.So, if we take any one job type as a reference point, the processor is alternatively busy handling that job type and absent doing work elsewhere [12].
Server vacation has been used successfully to model polling systems [13], multiclass priority scheduling systems [18], maintenance models [17] and processor failure [2].For a survey on the use of vacation models, see Doshi [7].
Most of the papers on vacation models assume the source population of the queueing system to be infinite.However, Trivdei et al. [11] have studied several vacation models in which they have taken the population to be finite.They assume an exponentially distributed service time.This paper is an extension of their work to the case where the service times are generally distributed.
We obtain Markov regenerative stochastic Petri net (MRSPN) representations of the M/G/1/N queueing systems with server vacations under a limited service discipline.For an excellent introduction to the theory of M/G/1 queueing systems, see Gross and Harris [9] and Takagi [14].The book by Takagi also contains information about queueing systems with server vacations.We consider three types of vacation schemes, namely the multiple vacation scheme, the single vacation scheme and the hybrid vacation scheme.We also show how the representations may be extended to the case where the population is infinite, but the system has a finite capacity.
The chief advantages of an MRSPN representation are: 1.The representation is aesthetically appealing and simple to understand, even for a layman.
2. Several other features of vacation queueing systems may be included in this representation, whose inclusion in other classical models would have led to analytically intractable solutions.
3. A considerable amount of research has been reported over the past two decades to develop software packages for stochastic Petri nets.As far as Markovian stochastic Petri nets are concerned, researchers have succeeded in developing the packages SPNP [15] and GreatSPN [5], which automatically generate the reachability graph and give efficient and reliable numerical solutions.The user only has to fit the model.For non-Markovian SPNs, several packages have been developed and a considerable amount of research is still ongoing in this direction.Two such packages are the WEBSPN [3] and SHARPE [16].
The rest of the paper is organised as follows: In §2 we give a brief outline of the theory of MRSPNs.In §3 we present MRSPN representations for the M/G/1 vacation queueing systems with a limited service discipline.We use the notation M/G/1/V l to denote such queueing systems in what follows.We consider the multiple, single vacation schemes and vacation schemes with a hybrid policy.We take up both the cases of a finite population and an infinite population with a finite system capacity.In §4 we illustrate the representation given in §3 with the help of simple numerical examples.Finally, in §5 we present a few concluding remarks.

Markov regenerative stochastic Petri nets
For the basics of the Petri net formalism, see Ajmone-Marsen [1] and German [8].We begin with the theory of MRSPNs in this section.
Transitions with an exponentially distributed firing time are referred to as EXP transitions.
Transitions with a generally distributed firing time are referred to as GEN transitions.
If the Petri net contains GEN transitions, the underlying marking process M (t) is not a continuous time Markov chain and it cannot be studied assuming the Markov chain theory.
As originally observed by Choi et al. [6], the underlying marking process is a Markov regenerative process under the conditions that: 1. there is at most one GEN transition enabled in any marking, and 2. the firing time value is sampled at the time the transition is enabled; it cannot change until the transition either fires or is disabled; a new firing time is obtained when the transition becomes enabled again.
Definition 1 A Markov regenerative process (MRGP) is a continuous-time discrete state stochastic process with an embedded sequence of regenerative time points (RTPs), at which the process enjoys the Markov property.
Definition 2 A Markov regenerative stochastic Petri net (MRSPN) is a stochastic Petri net in which the underlying marking process is a MRGP.
MRSPNs may be studied by using the theory of Markov regenerative processes.Based on the concept of memory in a stochastic Petri net, the regeneration time points may be defined.The memory of a transition in a stochastic Petri net is given by a pair of random variables (a, i).Here a is the age variable which keeps track of the amount of time elapsed since the enabling of the transition.The resampling indicator i is set equal to one whenever the firing time of the transition is to be resampled.Otherwise it is set equal to zero.Thus, a transition has non zero memory at any instant of time only if at least one of the variables a or i is non-zero.In the pre-emptive repeat different policy considered in this paper, the resampling variable and the age variable are reset as zero each time the transition is either disabled or is fired.
Definition 3 A Regeneration Time Point in the marking process {M(t) : t ≥ 0} is an instant of time at which all the transitions in the Petri net have zero memory.
Because of the memoryless property of the MRGP at the RTPs, the analysis of a MRSPN can be split into independant subproblems wherein the subordinated processes between any two consecutive RTPs are examined.The probability functions necessary for the analysis of a MRSPN are commonly referred to as the global and local kernels.
The global kernel is defined as where M (T 1 +) represents the marking immediately after the end of the regeneration cycle, where M (0) is the initial marking and where T 1 is the duration of the regeneration cycle.The local kernel is defined as In order to obtain the steady state probabilities of the MRSPN, we need to obtain the steady state probabilities of the embedded Markov chain (EMC) at the RTPs.These are obtained as the solution of v = vP , where P = K(∞) is the matrix whose ij th entry is Define µ j = k∈Ω α jk .Then the steady state probabilities P j of the MRSPN are given by the expressions For more details, the reader is referred to Choi et al. [6].
3 MRSPN representation of the M/G/1/V l queueing system In this section we consider a queueing system with a single service facility which becomes unavailable from time to time.Arrivals to the system are in accordance with a Poisson process with a parameter λ.The service times of the individual customers are independent identically distributed (i.i.d.) random variables with a distribution function B(t).The arrival process is assumed to be independant of the service mechanism.We assume that β(s) is the Laplace Stieltjes transform of the service time distribution B(t).The expected service time of a customer in the system is given by µ = −β (0).The duration of the server vacation is assumed to be exponentially distributed with a parameter γ.

A multiple vacation scheme
Under the multiple vacation scheme, the server goes on a vacation whenever there are no customers waiting for service in the system, that is, the system is empty.However, it is also allowed to go away after completion of a fixed number of, say k services, even if there are customers present in the system.

Systems with a finite population.
We first consider the case of a finite population of size N .The MRSPN representation of the system is given in Figure 1.
There are five places P I , P B , P s , P D and P v , and five transitions t a , t v , t s , S and S 1 .The transition t a whose firing indicates the arrival of a customer from the population P I to the system is an EXP transition.The transition t v whose firing represents the end of the server vacation is also an EXP transition.The transition t s whose firing represents the completion of a service is a GEN transition.The transitions S and S 1 are immediate transitions which fire in zero time.
The place P D is used to keep a record of the number of service completions.As soon as k services are completed, i.e. k tokens are deposited in P D , further service is suspended in the system.This is represented by an inhibitor arc of multiplicity k from the place P D to the GEN transition t s .The server then proceeds on vacation which is effected by the immediate transition S 1 .An inhibitor arc from P B to the transition S prevents the server from proceeding on a vacation when there is even a single customer in the system.The firing of S flushes out all the tokens in P D , so that the system can start afresh, after the return of the server from the vacation.The marking represents the state of the system at any instant of time t, where M (P I ), M (P B ), M (P s ), M (P D ), M (P v ) are the number of tokens in P I , P B , P s , P D , P v respectively, at time t.Thus, the initial marking of the system, at time t = 0, is M (0) = (N, 0, 1, 0, 0).This however, is a vanishing marking and is merged with its successor marking.Thus, M (0) = (N, 0, 0, 0, 1).
When the number of tokens in P I is N − r, the number of tokens in P B is r.Again, when the number of tokens in P s is 1, the number of tokens in P v is 0. It is therefore enough to signify a marking by the number of tokens in the places P B , P s and P D .Hence represents the marking at any instant of time t.The state space of the system is Since there is only one GEN transition in the model, the associated marking process is an MRGP.The transitions t a , t s and t v are not conflicting.Thus, the question of pre-emption does not arise.The results of Choi et al. [6] may therefore be used to obtain the transient and steady state probabilities of the system.
2. If, at the n th RTP T n , the system is in the state (r, 0, 0), the GEN transition t s , is not enabled.Therefore T n+1 is the time at which either of the two EXP transitions t a or t v fires.
3. If, at the n th RTP T n , the system is in the state (r, 1, n) with 1 ≤ r ≤ N , 0 ≤ n ≤ k − 1, the GEN transition t s is enabled.The next RTP T n+1 is the time at which t s fires.
In this case, the subordinated process is a continuous time Markov chain (CTMC) with state space Ω(r, 1, The transient and steady state solution of this CTMC are obtained by determining the rate matrix, which is known as the infinitisimal generator (see Medhi [10, p. 195]).The infinitesimal generator of the subordinated CTMC is given by Q where The expressions for the local kernel and the global kernel may now be obtained.
The performance measures of the system are the effective arrival rate (denoted by λ eff ), the expected number of customers in the system (denoted by L s ), the expected waiting time of a customer in the system (denoted by W s ), the expected number of vacations per unit time (denoted by µ v ) and the fraction of busy time during the interval [0, t] of the server (denoted by W t ).It follows that P (r,1,n) + N P (0,0,0) and Furthermore by Little's formula, (see, for example, Gross and Harris [9, p. 79]), P (r,0,0) and

Systems with a finite capacity
In this section, we assume that the population may be infinite, but that the capacity, denoted by N , is finite.Figure 1 may also be used as the MRSPN representation of this queueing model, except for two changes: 1.An inhibitor arc from the place P B to the arrival transition t a of multiplicity N prevents further arrivals when there are N customers in the system.
2. Since the source population is infinite, the arrival rate is no longer marking dependant.Thus, the symbol # placed next to t a in Figure 1 is now not necessary.
Again, since there is only one GEN transition and no pre-emption occurs, all the conditions necessary for the marking process {M (t) : t ≥ 0} to be a MRGP, are satisfied.The RTPs and state space Ω are as in §3.1.1.The subordinated process in a regeneration cycle starting in the state (r, 1, n) with 1 ≤ r ≤ N , 0 ≤ n ≤ k − 1 is a CTMC with state space Ω(r, 1, n), where Ω(r, 1, n) is as defined in §3.1.1.Here, the infinitesimal generator is where where 1 ≤ i ≤ N − r and the (N − r + 1) th row contains only zero's.Expressions for the local and global kernels may now be obtained.

Local kernel
When the regeneration cycle starts in the state (r, 0, 0) with 0 ≤ r < N , When the regeneration cycle starts in the state (N, 0, 0), When the regeneration cycle starts in the state (r, 1, n), the expressions for the local kernel are as in §3.1.1.
As in §3.1.1,if P (r,0,0) for 0 ≤ r ≤ N and P (r,1,n) for 1 ≤ r ≤ N , 0 ≤ n ≤ k − 1 denote steady state probabilities of the system, the expressions for the performance measures of the system are the same as in §3.1.1,except that the effective arrival rate is now given by

A single vacation scheme
In this section, it is assumed that the server goes on a vacation if the system is empty, provided that at least one service completion has occurred.However, if the server has completed a fixed number, say k services, it can proceed on a vacation even if the system is not empty.The MRSPN representation of the system is given in Figure 2.There is an additional place P C in this model to record the fact that the server has completed at least one service before going away on a vacation.The immediate transition S 2 , when enabled, places a token in P C , thus keeping a record of the first service completion.
An inhibitor arc from P C to S 2 ensures that this transition is not enabled again.The other services are recorded in P D .The immediate transition S whose enabling allows the server to proceed on vacation when the system is empty, now requires a input arc from P C .The multiplicity of the input arc from P D to S 1 is k − 1.The enabling of S 1 also requires an input arc from P C .An inhibitor input arc from P D to t s of multiplicity k − 1 prevents the enabling of t s when there are k − 1 tokens in P D .That is, a total of k service completions means that the service is terminated for some time.A zig-zag arc from P D to S flushes out all the tokens from P D when S fires (including when there are zero tokens in P D ).As in the previous section, the state of the system at any instant of time is given by where M (P B ), M (P s ), M (P C ) and M (P D ) denote the number of tokens in the places P B , P s , P C , P D respectively.The state space of the system is given by The RTPs are defined as follows: 1. T 0 = 0.
2. If at the n th RTP T n , the process is in the state (r, 0, 0, 0) with 0 ≤ r ≤ N , the GEN transition t s is not enabled.The next RTP T n+1 is the time at which either of the two EXP transitions t a or t v fires.
3. If at T n , the process is in the state (0, 1, 0, 0), the next RTP is the time at which t a fires.
4. If at T n , the process is in the state (r, 1, s, n) where 1 ≤ r ≤ N , 0 ≤ n ≤ k − 2 and s is either 0 or 1, the GEN transition t s is enabled.The next regeneration time point is the time at which t s fires.
In this case, the subordinated process is a CTMC with state space is as in the previous sections.The expressions for the local and global kernels may now be obtained.

A hybrid scheme
In this section it is assumed that the server can proceed on a vacation after the completion of a fixed number of services, as in the previous sections.However, if the system is empty, it can proceed on a vacation after waiting for a random period of time.This random amount of time is assumed to be exponentially distributed with parameter α.The MRSPN representation of the system is given in Figure 3.
: MRSPN representation of the M/G/1/V l queueing system with a hybrid scheme.
The places P I , P B , P s , P v , and the transitions t a , t s , t v have the same meaning as in the §3.1 and §3.2.The place P C keeps a record of the number of service completions.The transition t h allows the server to proceed on a vacation when the system is empty.The inhibitor arc from P B to t h prevents enabling of t h when the system is not empty.A zig-zag arc from P C to t h flushes out all the tokens in P C (including zero), when t h fires.The immediate transition S is enabled, when there are k tokens in P C .As in the previous section, there is an inhibitor arc of multiplicity k from P C to t s .As before, the marking M (t) at any instant of time t is specified by the number of tokens in P B , P s and P C .
Since there is only one GEN transition t s , which does not conflict with any of the EXP transitions t a , t h and t v , the underlying marking process is a MRGP.The RTPs may be defined as in the previous section.The state space of the system is given by The subordinated processes, which start in the state (r, 1, n) with 1 ≤ r ≤ N and 0 The infinitesimal generator is the same as that in the previous section.The expressions for the local and global kernels may now be obtained.

Local kernel
When the regeneration cycle starts in the state (r, 0, 0) with 0 ≤ r ≤ N , When the regeneration cycle starts in the state (0, 1, n) with 0 ≤ n ≤ k − 1, When the regeneration cycle starts in the states (r, 1, n) with 1 ≤ r ≤ N and 0

Numerical Examples
In this section we present three numerical examples to indicate how the MRSPN representations of the queueing systems considered in §3 may be used to obtain the performance measures of the system.In all three examples we use N = 3.

Limited service discipline with multiple vacation scheme
In this example we consider the vacation queueing system with a limited service discipline and a multiple vacation scheme.The reachability graph is given in Figure 4.This graph contains ten tangible markings {M j : 1 ≤ j ≤ 10} and five vanishing markings.After removing the vanishing markings, the state space of the process is given by Ω = {i : 1 ≤ i ≤ 10} where i stands for the marking M i .The local kernel is given by where A is a diagonal matrix of order 4 with e −[(4−i)λ+γ]t in the i th row and where B = [b ij ] is an upper triangular matrix of order 3 containing the elements The global kernel is given by where G is a diagonal matrix of order 3 containing as the element in the i th row.Furthermore H 1 = [h ij ] is a matrix of order 3 with only two non zero elements, namely Finally, Using the formulae in §2, the steady state probabilities P 1 , . . ., P 10 may be obtained as In the above, where µ is the expected value of the service distribution.The same reachability graph may be used to obtain the steady state probabilities of the queueing system when the population is infinite and the system capacity is three.The values for the steady state probabilities in both the finite population and finite capacity cases are given in Table 1.A deterministic service time µ is used.It may be seen from the table that for these values of µ and γ, the minimum probability is P 1 , corresponding to M 1 (0, 0, 0).Likewise, the maximum probability is P 4 , corresponding to the state M 4 (3, 0, 0).
4.2 Performance measures of the M/G/1/3/V l queueing systems under three different vacation schemes In this example, we compute some performance measures of the M/G/1/3 queueing system under a limited service discipline with k = 2.The values of the performance measures for the multiple, single and hybrid vacation schemes are given in Table 2.A deterministic service time distribution wirh service time µ is assumed.Also, the values λ = 1.0 and α = 3.0 were used.
From Table 2 it may be seen that for fixed values of µ and γ, the mean response time W s is the smallest in the case of a hybrid vacation scheme.The mean response time is smaller in the single vacation scheme than in the multiple vacation scheme.Similarly µ v , the number of vacations per unit time is smallest in the hybrid case.Consequently, the fraction of busy time of the server W t in the interval [0, t] is the largest in the hybrid scheme.The single vacation scheme is better than the multiple vacation scheme from the customer's point of view.

4.3
Mean response time for the M/G/1/3/V l queueing system with three types of service time distributions In this example, the mean response time associated with the three types of vacation schemes and service time distributions are compared for three types of service time distributions: a deterministic case, a three-phase Erlang distribution, and a hyper-exponential distribution.The expected service time in all three distributions is assumed to have the same value µ.Values of the mean response time for different values of µ, for the three distributions and for the three types of vacation schemes, are given in Table 3.
From Table 3, it may be seen that, as far as the single vacation and the multiple vacation schemes are concerned, the deterministic distribution gives the minimum values for the

Figure 2 :
Figure 2: MRSPN model of the single vacation M/G/1/N/V l queueing system.

Table 1 :
Steady state probabilities for the finite population and finite capacity M/G/1/3

Table 2 :
Performance measures for the M/G/1/3/V l queueing systems with multiple, single and hybrid schemes.